(** A multi-way branch is composed of a list of (key, action) pairs,
  plus a default action.  *)

Definition table : Type := list (Z * nat).Fixpoint switch_target (n: Z) (dfl: nat) (cases: table)
                       {struct cases} : nat :=
  match cases with
  | nil => dfl
  | (key, action) :: rem =>
      if zeq n key then action else switch_target n dfl rem
  end.Inductive switch_argument: bool -> val -> Z -> Prop :=
  | switch_argument_32: forall i,
      switch_argument false (Vint i) (Int.unsigned i)
  | switch_argument_64: forall i,
      switch_argument true (Vlong i) (Int64.unsigned i).Inductive switch_argument: bool -> val -> Z -> Prop :=
  | switch_argument_32: forall i,
      switch_argument false (Vint i) (Int.unsigned i)
  | switch_argument_64: forall i,
      switch_argument true (Vlong i) (Int64.unsigned i).Inductive switch_argument: bool -> val -> Z -> Prop :=
  | switch_argument_32: forall i,
      switch_argument false (Vint i) (Int.unsigned i)
  | switch_argument_64: forall i,
      switch_argument true (Vlong i) (Int64.unsigned i).(** Multi-way branches are translated to comparison trees.
    Each node of the tree performs either
- an equality against one of the keys;
- or a "less than" test against one of the keys;
- or a computed branch (jump table) against a range of key values. *)

Inductive comptree : Type :=
  | CTaction (act: nat)
  | CTifeq (key: Z) (act: nat) (cne: comptree)
  | CTiflt (key: Z) (clt: comptree) (cge: comptree)
  | CTjumptable (ofs: Z) (sz: Z) (acts: list nat) (cother: comptree).(** Multi-way branches are translated to comparison trees.
    Each node of the tree performs either
- an equality against one of the keys;
- or a "less than" test against one of the keys;
- or a computed branch (jump table) against a range of key values. *)

Inductive comptree : Type :=
  | CTaction (act: nat)
  | CTifeq (key: Z) (act: nat) (cne: comptree)
  | CTiflt (key: Z) (clt: comptree) (cge: comptree)
  | CTjumptable (ofs: Z) (sz: Z) (acts: list nat) (cother: comptree).(** Multi-way branches are translated to comparison trees.
    Each node of the tree performs either
- an equality against one of the keys;
- or a "less than" test against one of the keys;
- or a computed branch (jump table) against a range of key values. *)

Inductive comptree : Type :=
  | CTaction (act: nat)
  | CTifeq (key: Z) (act: nat) (cne: comptree)
  | CTiflt (key: Z) (clt: comptree) (cge: comptree)
  | CTjumptable (ofs: Z) (sz: Z) (acts: list nat) (cother: comptree).(** Multi-way branches are translated to comparison trees.
    Each node of the tree performs either
- an equality against one of the keys;
- or a "less than" test against one of the keys;
- or a computed branch (jump table) against a range of key values. *)

Inductive comptree : Type :=
  | CTaction (act: nat)
  | CTifeq (key: Z) (act: nat) (cne: comptree)
  | CTiflt (key: Z) (clt: comptree) (cge: comptree)
  | CTjumptable (ofs: Z) (sz: Z) (acts: list nat) (cother: comptree).(** Multi-way branches are translated to comparison trees.
    Each node of the tree performs either
- an equality against one of the keys;
- or a "less than" test against one of the keys;
- or a computed branch (jump table) against a range of key values. *)

Inductive comptree : Type :=
  | CTaction (act: nat)
  | CTifeq (key: Z) (act: nat) (cne: comptree)
  | CTiflt (key: Z) (clt: comptree) (cge: comptree)
  | CTjumptable (ofs: Z) (sz: Z) (acts: list nat) (cother: comptree).Fixpoint comptree_match (n: Z) (t: comptree) {struct t}: option nat :=
  match t with
  | CTaction act => Some act
  | CTifeq key act t' =>
      if zeq n key then Some act else comptree_match n t'
  | CTiflt key t1 t2 =>
      if zlt n key then comptree_match n t1 else comptree_match n t2
  | CTjumptable ofs sz tbl t' =>
      let delta := (n - ofs) mod modulus in
      if zlt delta sz
      then list_nth_z tbl (delta mod Int.modulus)
      else comptree_match n t'
  end.(** The translation from a table to a comparison tree is performed
  by untrusted Caml code (function [compile_switch] in
  file [RTLgenaux.ml]).  In Coq, we validate a posteriori the
  result of this function.  In other terms, we now develop
  and prove correct Coq functions that take a table and a comparison
  tree, and check that their semantics are equivalent. *)

Fixpoint split_lt (pivot: Z) (cases: table)
                  {struct cases} : table * table :=
  match cases with
  | nil => (nil, nil)
  | (key, act) :: rem =>
      let (l, r) := split_lt pivot rem in
      if zlt key pivot
      then ((key, act) :: l, r)
      else (l, (key, act) :: r)
  end.Fixpoint split_eq (pivot: Z) (cases: table)
                  {struct cases} : option nat * table :=
  match cases with
  | nil => (None, nil)
  | (key, act) :: rem =>
      let (same, others) := split_eq pivot rem in
      if zeq key pivot
      then (Some act, others)
      else (same, (key, act) :: others)
  end.Fixpoint split_between (dfl: nat) (ofs sz: Z) (cases: table)
                       {struct cases} : ZMap.t nat * table :=
  match cases with
  | nil => (ZMap.init dfl, nil)
  | (key, act) :: rem =>
      let (inside, outside) := split_between dfl ofs sz rem in
      if zlt ((key - ofs) mod modulus) sz
      then (ZMap.set key act inside, outside)
      else (inside, (key, act) :: outside)
  end.Definition refine_low_bound (v lo: Z) :=
  if zeq v lo then lo + 1 else lo.Definition refine_high_bound (v hi: Z) :=
  if zeq v hi then hi - 1 else hi.Fixpoint validate_jumptable (cases: ZMap.t nat)
                            (tbl: list nat) (n: Z) {struct tbl} : bool :=
  match tbl with
  | nil => true
  | act :: rem =>
      Nat.eqb act (ZMap.get n cases)
      && validate_jumptable cases rem (Z.succ n)
  end.Fixpoint validate (default: nat) (cases: table) (t: comptree)
                  (lo hi: Z) {struct t} : bool :=
  match t with
  | CTaction act =>
      match cases with
      | nil =>
          Nat.eqb act default
      | (key1, act1) :: _ =>
          zeq key1 lo && zeq lo hi && Nat.eqb act act1
      end
  | CTifeq pivot act t' =>
      zle 0 pivot && zlt pivot modulus &&
      match split_eq pivot cases with
      | (None, _) =>
          false
      | (Some act', others) =>
          Nat.eqb act act'
          && validate default others t'
                      (refine_low_bound pivot lo)
                      (refine_high_bound pivot hi)
      end
  | CTiflt pivot t1 t2 =>
      zle 0 pivot && zlt pivot modulus &&
      match split_lt pivot cases with
      | (lcases, rcases) =>
          validate default lcases t1 lo (pivot - 1)
          && validate default rcases t2 pivot hi
      end
  | CTjumptable ofs sz tbl t' =>
      let tbl_len := list_length_z tbl in
      zle 0 ofs && zlt ofs modulus &&
      zle 0 sz && zlt sz modulus &&
      zle (ofs + sz) modulus && zle sz tbl_len && zlt sz Int.modulus &&
      match split_between default ofs sz cases with
      | (inside, outside) =>
          validate_jumptable inside tbl ofs
          && validate default outside t' lo hi
     end
  end.Definition validate_switch (default: nat) (cases: table) (t: comptree) :=
  validate default cases t 0 (modulus - 1).(** Structural properties checked by validation *)

Inductive wf_comptree: comptree -> Prop :=
  | wf_action: forall act,
      wf_comptree (CTaction act)
  | wf_ifeq: forall key act cne,
      0 <= key < modulus -> wf_comptree cne -> wf_comptree (CTifeq key act cne)
  | wf_iflt: forall key clt cge,
      0 <= key < modulus -> wf_comptree clt -> wf_comptree cge -> wf_comptree (CTiflt key clt cge)
  | wf_jumptable: forall ofs sz acts cother,
      0 <= ofs < modulus -> 0 <= sz < modulus ->
      wf_comptree cother ->
      wf_comptree (CTjumptable ofs sz acts cother).(** Structural properties checked by validation *)

Inductive wf_comptree: comptree -> Prop :=
  | wf_action: forall act,
      wf_comptree (CTaction act)
  | wf_ifeq: forall key act cne,
      0 <= key < modulus -> wf_comptree cne -> wf_comptree (CTifeq key act cne)
  | wf_iflt: forall key clt cge,
      0 <= key < modulus -> wf_comptree clt -> wf_comptree cge -> wf_comptree (CTiflt key clt cge)
  | wf_jumptable: forall ofs sz acts cother,
      0 <= ofs < modulus -> 0 <= sz < modulus ->
      wf_comptree cother ->
      wf_comptree (CTjumptable ofs sz acts cother).(** Structural properties checked by validation *)

Inductive wf_comptree: comptree -> Prop :=
  | wf_action: forall act,
      wf_comptree (CTaction act)
  | wf_ifeq: forall key act cne,
      0 <= key < modulus -> wf_comptree cne -> wf_comptree (CTifeq key act cne)
  | wf_iflt: forall key clt cge,
      0 <= key < modulus -> wf_comptree clt -> wf_comptree cge -> wf_comptree (CTiflt key clt cge)
  | wf_jumptable: forall ofs sz acts cother,
      0 <= ofs < modulus -> 0 <= sz < modulus ->
      wf_comptree cother ->
      wf_comptree (CTjumptable ofs sz acts cother).(** Structural properties checked by validation *)

Inductive wf_comptree: comptree -> Prop :=
  | wf_action: forall act,
      wf_comptree (CTaction act)
  | wf_ifeq: forall key act cne,
      0 <= key < modulus -> wf_comptree cne -> wf_comptree (CTifeq key act cne)
  | wf_iflt: forall key clt cge,
      0 <= key < modulus -> wf_comptree clt -> wf_comptree cge -> wf_comptree (CTiflt key clt cge)
  | wf_jumptable: forall ofs sz acts cother,
      0 <= ofs < modulus -> 0 <= sz < modulus ->
      wf_comptree cother ->
      wf_comptree (CTjumptable ofs sz acts cother).(** Structural properties checked by validation *)

Inductive wf_comptree: comptree -> Prop :=
  | wf_action: forall act,
      wf_comptree (CTaction act)
  | wf_ifeq: forall key act cne,
      0 <= key < modulus -> wf_comptree cne -> wf_comptree (CTifeq key act cne)
  | wf_iflt: forall key clt cge,
      0 <= key < modulus -> wf_comptree clt -> wf_comptree cge -> wf_comptree (CTiflt key clt cge)
  | wf_jumptable: forall ofs sz acts cother,
      0 <= ofs < modulus -> 0 <= sz < modulus ->
      wf_comptree cother ->
      wf_comptree (CTjumptable ofs sz acts cother).

Lemma split_between_prop: forall v default ofs sz cases inside outside, split_between default ofs sz cases = (inside, outside) -> switch_target v default cases = (if zlt ((v - ofs) mod modulus) sz then ZMap.get v inside else switch_target v default outside).