From 73fd1921f90824e481c0f8e0c52003d37af5a2e5 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Wed, 10 May 2023 19:22:32 +0200
Subject: Cleanup divDefProtoScript

---
 backends/hol4/divDefProtoScript.sml | 525 +-----------------------------------
 1 file changed, 2 insertions(+), 523 deletions(-)

(limited to 'backends/hol4/divDefProtoScript.sml')

diff --git a/backends/hol4/divDefProtoScript.sml b/backends/hol4/divDefProtoScript.sml
index 64d4b9f0..6280fa7e 100644
--- a/backends/hol4/divDefProtoScript.sml
+++ b/backends/hol4/divDefProtoScript.sml
@@ -101,7 +101,7 @@ Proof
 QED
 
 (* TODO: I think we can merge this with the theorem below *)
-Theorem fix_fixed_termination_rec_case:
+Theorem fix_fixed_termination_rec_case_aux:
   ∀x y n m.
   is_valid_fp_body f ⇒
   (∀g. f g x = g y) ⇒
@@ -128,7 +128,7 @@ Theorem fix_fixed_termination_rec_case:
 Proof
   rw [] >>
   imp_res_tac fix_fuel_mono_least >>
-  irule fix_fixed_termination_rec_case >>
+  irule fix_fixed_termination_rec_case_aux >>
   fs [] >>
   (* TODO: factorize *)
   qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
@@ -249,525 +249,4 @@ Proof
   simp_tac arith_ss []
 QED
 
-(************************** FAILED TESTS *)
-
-(*
-Datatype:
-  fresult = FReturn 'b | FFail error | FRec 'a
-End
-
-(* TODO: monad converter from fresult to result + rewrite rules *)
-val fres_to_res_def = Define ‘
-  fres_to_res (f : 'a -> 'b result) (r : ('a, 'b) fresult) : 'b result =
-    case r of
-    | FReturn x => Return x
-    | FFail e => Fail e
-    | FRec y => f y
-’
-
-val fixa_fuel_def = Define ‘
-  (fixa_fuel (0 : num) (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result = Diverge) ∧
-  (fixa_fuel (SUC n) (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result =
-    fres_to_res (fixa_fuel n f) (f x))
-’
-
-val fixa_fuel_P_def = Define ‘
-  fixa_fuel_P f x n = ~(is_diverge (fixa_fuel n f x))
-’
-
-val fixa_def = Define ‘
-  fixa (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result =
-    if (∃ n. fixa_fuel_P f x n) then fixa_fuel ($LEAST (fixa_fuel_P f x)) f x else Diverge
-’
-
-Theorem fixa_fuel_mono:
-  ∀n m. n <= m ⇒ (∀f x. fixa_fuel_P f x n ⇒ fixa_fuel n f x = fixa_fuel m f x)
-Proof
-  Induct_on ‘n’ >> rpt strip_tac
-  >- (fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def]) >>
-  Cases_on ‘m’ >- int_tac >> fs [] >>
-  last_x_assum imp_res_tac >>
-  fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def] >>
-  Cases_on ‘f x’ >> fs []
-QED
-
-(* TODO: remove ? *)
-Theorem fixa_fuel_P_least:
-  ∀f x n. fixa_fuel_P f x n ⇒ fixa_fuel_P f x ($LEAST (fixa_fuel_P f x))
-Proof
-  rw [] >>
-  (* Use the "fundamental" property about $LEAST *)
-  qspec_assume ‘fixa_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
-  (* Prove the premise *)
-  pop_assum sg_premise_tac >- metis_tac [] >>
-  rw []
-QED
-
-Theorem fixa_fuel_mono_least:
-  ∀n f x. fixa_fuel_P f x n ⇒ fixa_fuel n f x = fixa_fuel ($LEAST (fixa_fuel_P f x)) f x
-Proof
-  rw [] >>
-  (* Use the "fundamental" property about $LEAST *)
-  qspec_assume ‘fixa_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
-  (* Prove the premise *)
-  pop_assum sg_premise_tac >- metis_tac [] >>
-  (* Prove that $LEAST ... ≤ n *)
-  sg ‘n >= $LEAST (fixa_fuel_P f x)’ >-(
-    spose_not_then assume_tac >>
-    fs [not_ge_eq_lt]) >>
-  pure_once_rewrite_tac [EQ_SYM_EQ] >>
-  irule fixa_fuel_mono >>
-  fs []
-QED
-
-Theorem fixa_fixed_eq:
-  ∀f x. fixa f x = fres_to_res (fixa f) (f x)
-Proof
-  rw [fixa_def, fres_to_res_def]
-  >- (
-    (* Termination case *)
-    imp_res_tac fixa_fuel_P_least >>
-    Cases_on ‘$LEAST (fixa_fuel_P f x)’ >>
-    fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def] >>
-    Cases_on ‘f x’ >> fs [] >>
-    rw [] >> fs [] >>
-    irule fixa_fuel_mono_least >>
-    fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def]) >>
-  (* Divergence case *)
-  fs [fres_to_res_def] >>
-  sg ‘∀n. ~(fixa_fuel_P f x (SUC n))’ >- fs [] >>
-  Cases_on ‘f x’ >> fs [] >>
-  fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def]  >>
-  rw []
-QED
-
-val fix_fuel_def = Define ‘
-  (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
-  (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
-’
-
-val fix_fuel_P_def = Define ‘
-  fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
-’
-
-val fix_def = Define ‘
-  fix (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
-    if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
-’
-
-Theorem fixa_fuel_eq_fix_fuel:
-  ∀n (f : 'a -> ('a, 'b) fresult) (x : 'a).
-    fixa_fuel n f x = fix_fuel n (\g y. fres_to_res g (f y)) x
-Proof
-  Induct_on ‘n’ >> rw [fixa_fuel_def, fix_fuel_def, fres_to_res_def]
-QED
-
-Theorem fixa_fuel_P_eq_fix_fuel_P:
-  ∀(f : 'a -> ('a, 'b) fresult) (x : 'a).
-    fixa_fuel_P f x = fix_fuel_P (\g y. fres_to_res g (f y)) x
-Proof
-  rw [] >> irule EQ_EXT >> rw [] >>
-  qspecl_assume [‘x'’, ‘f’, ‘x’] fixa_fuel_eq_fix_fuel >>
-  fs [fixa_fuel_P_def, fix_fuel_P_def]  
-QED
-
-Theorem fixa_fuel_eq_fix_fuel:
-  ∀(f : 'a -> ('a, 'b) fresult) (x : 'a).
-    fixa f x = fix (\g y. fres_to_res g (f y)) x
-Proof
-  rw [fixa_def, fix_def, fixa_fuel_P_eq_fix_fuel_P, fixa_fuel_eq_fix_fuel]
-QED
-
-(*
-  The annoying thing is that to make this work, we need to rewrite the
-  body by swapping matches, which is not easy to automate.
-*)
-
-(*
-f x =
-  case _ of
-  | _ => Fail e
-  | _ => Return x
-  | _ => f y
-
-f x (g : ('a, 'b) result => 'c)
-
-f x (fres_to_res g) = f x g
-
-*)
-
-(*
-val fix_fuel_def = Define ‘
-  (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
-  (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
-’
-
-val fix_fuel_P_def = Define ‘
-  fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
-’
-*)
-
-
-(*
-val is_valid_FP_body_def = Define ‘
-  is_valid_FP_body (f : (('a, 'b) fresult -> 'c) -> ('a, 'b) fresult -> 'c) =
-    ∃ (fa: (('a, 'b) fresult -> ('a, 'b) fresult) -> ('a, 'b) fresult -> ('a, 'b) fresult).
-      ∀ (g : ('a, 'b) fresult -> 'c) x.
-        f g x = g (fa (\x. x) x)
-
-val fix_fuel_def = Define ‘
-  (fix_fuel (0 : num) (f : (('a, 'b) fresult -> 'b result) -> ('a, 'b) fresult -> 'b result) (x : ('a, 'b) fresult) : 'b result = Diverge) ∧
-  (fix_fuel (SUC n) (f : (('a, 'b) fresult -> 'b result) -> ('a, 'b) fresult -> 'b result) (x : ('a, 'b) fresult) : 'b result = f (fix_fuel n f) x)
-’
-
-val fix_fuel_P_def = Define ‘
-  fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
-’
-
-val fix_def = Define ‘
-  fix f x : 'b result =
-    if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
-
-
-Theorem fix_fuel_mono:
-  ∀f. is_valid_FP_body f ⇒
-  ∀n m. n <= m ⇒ (∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
-Proof
-  strip_tac >> strip_tac >>
-  Induct_on ‘n’ >> rpt strip_tac
-  >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-  Cases_on ‘m’ >- int_tac >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-  fs [is_valid_FP_body] >>
-  last_x_assum (qspec_assume ‘fix_fuel n f’) >>
-  fs []
-QED
-
-’*)
-
-(*
- * Test with a validity predicate which gives monotonicity.
-
-   TODO: not enough to prove the fixed-point equation.
- *)
-val fix_fuel_def = Define ‘
-  (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
-  (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
-’
-
-val fix_fuel_P_def = Define ‘
-  fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
-’
-
-val fix_def = Define ‘
-  fix (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
-    if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
-’
-
-Theorem fix_eq:
-  fix f = \x. if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
-Proof
-  irule EQ_EXT >> fs [fix_def]
-QED
-
-val is_valid_fp_body_def = Define ‘
-  is_valid_fp_body (f : ('a -> 'b result) -> 'a -> 'b result) =
-    ∀n m. n ≤ m ⇒ ∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x
-’
-
-Theorem fix_fuel_mono_least:
-  ∀f. is_valid_fp_body f ⇒
-    ∀n x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel ($LEAST (fix_fuel_P f x)) f x
-Proof
-  rw [] >>
-  (* Use the "fundamental" property about $LEAST *)
-  qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
-  (* Prove the premise *)
-  pop_assum sg_premise_tac >- metis_tac [] >> fs [] >>
-  (* Prove that $LEAST ... ≤ n *)
-  sg ‘n >= $LEAST (fix_fuel_P f x)’ >-(
-    spose_not_then assume_tac >>
-    fs [not_ge_eq_lt]) >>
-  pure_once_rewrite_tac [EQ_SYM_EQ] >>
-  (* Finish by using the monotonicity property *)
-  fs [is_valid_fp_body_def]
-QED
-
-
-(* TODO: remove ? *)
-Theorem fix_fuel_P_least:
-  ∀f x n. fix_fuel_P f x n ⇒ fix_fuel_P f x ($LEAST (fix_fuel_P f x))
-Proof
-  rw [] >>
-  (* Use the "fundamental" property about $LEAST *)
-  qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
-  (* Prove the premise *)
-  pop_assum sg_premise_tac >- metis_tac [] >>
-  rw []
-QED
-
-(*Theorem test:
-  fix_fuel_P f x (SUC n) ⇒
-  (∀m. n ≤ m ⇒ f (fix_fuel n f) x = f (fix_fuel m f) x) ⇒
-  f (fix_fuel n f) x = f (fix f) x
-Proof
-  rw [] >>
-  spose_not_then assume_tac >>
-  fs [fix_eq]
-*)
-Theorem fix_fixed_eq:
-  ∀f. is_valid_fp_body f ⇒ ∀x. fix f x = f (fix f) x
-Proof
-  rw [fix_def] >> CASE_TAC >> fs []
-  >- (
-    (* Termination case *)
-    sg ‘$LEAST (fix_fuel_P f x) <= n’ >- cheat >>
-    imp_res_tac fix_fuel_P_least >>
-    Cases_on ‘$LEAST (fix_fuel_P f x)’ >>
-    fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-    Cases_on ‘n’ >>
-    fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-    (* Use the validity assumption *)
-    fs [is_valid_fp_body_def] >>
-  
-(*    last_x_assum imp_res_tac >> *)
-    last_assum (qspecl_assume [‘SUC n'’, ‘SUC n''’]) >> fs [] >>
-    pop_assum imp_res_tac >>
-    pop_assum (qspec_assume ‘x’) >>
-    pop_assum sg_premise_tac >- fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-    
-    fs [fix_fuel_def] >>
-
-    
-    fs [fix_fuel_P_def] >>
-    rfs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-    
-    fs [fix_eq] >>
-
-    Cases_on ‘fix_fuel n' f x’ >> fs [] >>
-    last_assum (qspecl_then [‘n'’, ‘’
-    f (fix_fuel n' f) x >>
-
-    Cases_on ‘f x’ >> fs [] >>
-    rw [] >> fs [] >>
-    irule fix_fuel_mono_least >>
-    fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
-  (* Divergence case *)
-  fs [fres_to_res_def] >>
-  sg ‘∀n. ~(fix_fuel_P f x (SUC n))’ >- fs [] >>
-  Cases_on ‘f x’ >> fs [] >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]  >>
-  rw []
-QED
-
-
-(* A slightly weaker and more precise validity criteria.
-
-   We simply extracted the part of the inductive proof that we can't prove without
-   knowing ‘f’.
- *)
-val is_valid_fp_body_weak_def = Define ‘
-   is_valid_fp_body_weak f =
-   ∀n m x.
-   ((∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x) ⇒
-    n ≤ m ⇒
-    fix_fuel_P f x (SUC n) ⇒
-    fix_fuel (SUC n) f x = fix_fuel (SUC m) f x)
-’
-
-Theorem is_valid_fp_body_weak_imp_is_valid:
-  ∀f. is_valid_fp_body_weak f ⇒ is_valid_fp_body f
-Proof
-  rpt strip_tac >> fs [is_valid_fp_body_def] >>
-  Induct_on ‘n’ >- fs [fix_fuel_P_def, fix_fuel_def, is_diverge_def] >>
-  Cases_on ‘m’ >> fs [] >>
-  fs [is_valid_fp_body_weak_def]
-QED
-
-(* Testing on an example *)
-Datatype:
-  list_t =
-    ListCons 't list_t
-  | ListNil
-End
-
-val nth_body_def = Define ‘
-  nth_body (f : ('t list_t # u32) -> 't result) (x : ('t list_t # u32)) : 't result =
-    let (ls, i) = x in
-    case ls of
-    | ListCons x tl =>
-      if u32_to_int i = (0:int)
-      then (Return x)
-      else
-        do
-        i0 <- u32_sub i (int_to_u32 1);
-        f (tl, i0)
-        od
-    | ListNil => (Fail Failure)
-’ 
-
-(* The general proof of is_valid_fp_body. We isolate a more precise property below. *)
-Theorem nth_body_is_valid:
-  is_valid_fp_body nth_body
-Proof
-  pure_rewrite_tac [is_valid_fp_body_def] >>
-  Induct_on ‘n’ >- fs [fix_fuel_P_def, fix_fuel_def, is_diverge_def] >>
-  Cases_on ‘m’ >- fs [] >>
-  rpt strip_tac >>
-  (* TODO: here *)
-  last_assum imp_res_tac >>
-  (* TODO: here? *)
-  fs [fix_fuel_def, nth_body_def, fix_fuel_P_def, is_diverge_def, bind_def] >>
-  Cases_on ‘x’ >> fs [] >> (* TODO: automate this *)
-  (* Explore all paths *)
-  Cases_on ‘q’ >> fs [] >>
-  Cases_on ‘u32_to_int r = 0’ >> fs [] >>
-  Cases_on ‘u32_sub r (int_to_u32 1)’ >> fs []
-QED
-
-
-Theorem nth_body_is_valid_weak:
-  is_valid_fp_body_weak nth_body
-Proof
-  pure_rewrite_tac [is_valid_fp_body_weak_def] >>
-  rpt strip_tac >>
-  (* TODO: automate this - we may need to destruct a variable number of times *)
-  Cases_on ‘x’ >> fs [] >>
-  (* Expand all *)
-  fs [fix_fuel_def, nth_body_def, fix_fuel_P_def, is_diverge_def, bind_def] >>
-  (* Explore all paths *)
-  Cases_on ‘q’ >> fs [] >>
-  Cases_on ‘u32_to_int r = 0’ >> fs [] >>
-  Cases_on ‘u32_sub r (int_to_u32 1)’ >> fs []
-QED
-
-(*
- * Test with a more general validity predicate.
- *)
-(* We need to control the way calls to the continuation are performed, so
-   that we can inspect the inputs (otherwise we can't prove anything).
- *)
-val is_valid_FP_body_def = Define ‘
-  is_valid_FP_body (f : (('a, 'b) fresult -> 'c) -> 'a -> 'c) =
-    ∃ (fa: (('a, 'b) fresult -> ('a, 'b) fresult) -> 'a -> ('a, 'b) fresult).
-      ∀ (g : ('a, 'b) fresult -> 'c) x.
-        f g x = g (fa (\x. x) x)
-’
-
-
-val fix_fuel_def = Define ‘
-  (fix_fuel (0 : num) (f : (('a, 'b) fresult -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
-  (fix_fuel (SUC n) (f : (('a, 'b) fresult -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
-    f (fres_to_res (fix_fuel n f)) x)
-’
-
-val fix_fuel_P_def = Define ‘
-  fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
-’
-
-val fix_def = Define ‘
-  fix f x : 'b result =
-    if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
-’
-
-Theorem fix_fuel_mono:
-  ∀f. is_valid_FP_body f ⇒
-  ∀n m. n <= m ⇒ (∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
-Proof
-  strip_tac >> strip_tac >>
-  Induct_on ‘n’ >> rpt strip_tac
-  >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def, fres_to_res_def] >>
-  Cases_on ‘m’ >- int_tac >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-  fs [is_valid_FP_body_def, fres_to_res_def] >>
-  CASE_TAC >>
-  last_x_assum (qspec_assume ‘fres_to_res (fix_fuel n f)’) >>
-  fs [fres_to_res_def]
-QED
-
-(* Tests *)
-
-
-
-Datatype:
-  list_t =
-    ListCons 't list_t
-  | ListNil
-End
-
-(*
-val def_qt = ‘
-  nth_mut_fwd (ls : 't list_t) (i : u32) : 't result =
-  case ls of
-  | ListCons x tl =>
-    if u32_to_int i = (0:int)
-    then Return x
-    else
-      do
-      i0 <- u32_sub i (int_to_u32 1);
-      nth_mut_fwd tl i0
-      od
-  | ListNil =>
-    Fail Failure
-’
-*)
-
-val nth_body_def = Define ‘
-  nth_body (f : ('t list_t # u32, 't) fresult -> 'c) (ls : 't list_t, i : u32) : 'c =
-    case ls of
-    | ListCons x tl =>
-      if u32_to_int i = (0:int)
-      then f (FReturn x)
-      else
-        do
-        i0 <- u32_sub i (int_to_u32 1);
-        f (FRec (tl, i0))
-        od
-    | ListNil => f (FFail Failure)
-’
-
-Theorem nth_body_is_valid_FP_body:
-  is_valid_FP_body nth_body
-Proof
-  fs [is_valid_FP_body_def] >>
-  exists_tac “nth_body” >>
-QED
-
-“nth_body”
-
-(* TODO: abbreviation for ‘(\g y. fres_to_res g (f y))’ *)
-Theorem fix_fixed_eq:
-  ∀f x. fix (\g y. fres_to_res g (f y)) x =
-     (f x)
-    fres_to_res (\g y. fres_to_res g (f y))
-
-
-
-(*
-TODO: can't prove that
-
-Theorem fix_fuel_mono:
-  ∀n m. n <= m ⇒ (∀f x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
-Proof
-  Induct_on ‘n’ >> rpt strip_tac
-  >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-  Cases_on ‘m’ >- int_tac >>
-  fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
-
-  sg ‘fix_fuel n f = fix_fuel n' f’ >>
-
-  last_x_assum imp_res_tac >>
-  pop_assum (qspecl_assume [‘x’, ‘f’]) >>
-
-Theorem fix_def_eq:
-  ∀f x. fix f x = f (fix f) x
-Proof
-  
-*)
-*)
-
-
 val _ = export_theory ()
-- 
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