Inductive tr_top: destination -> Csyntax.expr -> list statement -> expr -> list ident -> Prop :=
  | tr_top_val_val: forall v ty a tmp,
      typeof a = ty -> eval_expr ge e le m a v ->
      tr_top For_val (Csyntax.Eval v ty) nil a tmp
  | tr_top_base: forall dst r sl a tmp,
      tr_expr le dst r sl a tmp ->
      tr_top dst r sl a tmp.Inductive tr_top: destination -> Csyntax.expr -> list statement -> expr -> list ident -> Prop :=
  | tr_top_val_val: forall v ty a tmp,
      typeof a = ty -> eval_expr ge e le m a v ->
      tr_top For_val (Csyntax.Eval v ty) nil a tmp
  | tr_top_base: forall dst r sl a tmp,
      tr_expr le dst r sl a tmp ->
      tr_top dst r sl a tmp.Inductive tr_top: destination -> Csyntax.expr -> list statement -> expr -> list ident -> Prop :=
  | tr_top_val_val: forall v ty a tmp,
      typeof a = ty -> eval_expr ge e le m a v ->
      tr_top For_val (Csyntax.Eval v ty) nil a tmp
  | tr_top_base: forall dst r sl a tmp,
      tr_expr le dst r sl a tmp ->
      tr_top dst r sl a tmp.(** ** Translation of statements *)

Inductive tr_expression: Csyntax.expr -> statement -> expr -> Prop :=
  | tr_expression_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_expression r (makeseq sl) a.(** ** Translation of statements *)

Inductive tr_expression: Csyntax.expr -> statement -> expr -> Prop :=
  | tr_expression_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_expression r (makeseq sl) a.Inductive tr_expr_stmt: Csyntax.expr -> statement -> Prop :=
  | tr_expr_stmt_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_effects r sl a tmps) ->
      tr_expr_stmt r (makeseq sl).Inductive tr_expr_stmt: Csyntax.expr -> statement -> Prop :=
  | tr_expr_stmt_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_effects r sl a tmps) ->
      tr_expr_stmt r (makeseq sl).Inductive tr_if: Csyntax.expr -> statement -> statement -> statement -> Prop :=
  | tr_if_intro: forall r s1 s2 sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_if r s1 s2 (makeseq (sl ++ makeif a s1 s2 :: nil)).Inductive tr_if: Csyntax.expr -> statement -> statement -> statement -> Prop :=
  | tr_if_intro: forall r s1 s2 sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_if r s1 s2 (makeseq (sl ++ makeif a s1 s2 :: nil)).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).(** * Correctness proof with respect to the specification. *)

(** ** Properties of the monad *)

Remark bind_inversion:
  forall (A B: Type) (f: mon A) (g: A -> mon B) (y: B) (z1 z3: generator) I,
  bind f g z1 = Res y z3 I ->
  exists x, exists z2, exists I1, exists I2,
  f z1 = Res x z2 I1 /\ g x z2 = Res y z3 I2.Remark bind2_inversion:
  forall (A B Csyntax: Type) (f: mon (A*B)) (g: A -> B -> mon Csyntax) (y: Csyntax) (z1 z3: generator) I,
  bind2 f g z1 = Res y z3 I ->
  exists x1, exists x2, exists z2, exists I1, exists I2,
  f z1 = Res (x1,x2) z2 I1 /\ g x1 x2 z2 = Res y z3 I2.Ltac monadInv1 H :=
  match type of H with
  | (Res _ _ _ = Res _ _ _) =>
      inversion H; clear H; try subst
  | (@ret _ _ _ = Res _ _ _) =>
      inversion H; clear H; try subst
  | (@error _ _ _ = Res _ _ _) =>
      inversion H
  | (bind ?F ?G ?Z = Res ?X ?Z' ?I) =>
      let x := fresh "x" in (
      let z := fresh "z" in (
      let I1 := fresh "I" in (
      let I2 := fresh "I" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind_inversion _ _ F G X Z Z' I H) as [x [z [I1 [I2 [EQ1 EQ2]]]]];
      clear H;
      try (monadInv1 EQ2)))))))
   | (bind2 ?F ?G ?Z = Res ?X ?Z' ?I) =>
      let x := fresh "x" in (
      let y := fresh "y" in (
      let z := fresh "z" in (
      let I1 := fresh "I" in (
      let I2 := fresh "I" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind2_inversion _ _ _ F G X Z Z' I H) as [x [y [z [I1 [I2 [EQ1 EQ2]]]]]];
      clear H;
      try (monadInv1 EQ2))))))))
 end.Ltac monadInv H :=
  match type of H with
  | (@ret _ _ _ = Res _ _ _) => monadInv1 H
  | (@error _ _ _ = Res _ _ _) => monadInv1 H
  | (bind ?F ?G ?Z = Res ?X ?Z' ?I) => monadInv1 H
  | (bind2 ?F ?G ?Z = Res ?X ?Z' ?I) => monadInv1 H
  | (?F _ _ _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  end.(** ** Freshness and separation properties. *)

Definition within (id: ident) (g1 g2: generator) : Prop :=
  Ple (gen_next g1) id /\ Plt id (gen_next g2).

(* Specifications: *)
Lemma gensym_within: forall ty g1 id g2 I, gensym ty g1 = Res id g2 I -> within id g1 g2.
Lemma within_widen: forall id g1 g2 g1' g2', within id g1 g2 -> Ple (gen_next g1') (gen_next g1) -> Ple (gen_next g2) (gen_next g2') -> within id g1' g2'.