gallery: yet another implementation of McCarthy's 91 function

this time with a manually-optimized tail call
(and auto-deref applied to all references in this file)

examples/mccarthy.mlw

@@ -16,33 +16,45 @@ module McCarthy91
 
   (** {2 traditional recursive implementation} *)
 
-  let rec f91 (n:int) : int variant { 101-n }
-    ensures { result = spec n }
+  let rec f91 (n: int) : int
+    ensures { result = spec n } variant { 101 - n }
   = if n <= 100 then
       f91 (f91 (n + 11))
     else
       n - 10
 
+  (** {2 manually-optimized tail call} *)
+
+  let rec f91_tco (n0: int) : int
+    ensures { result = spec n0 } variant { 101 - n0 }
+  = let ref n = n0 in
+    while n <= 100 do
+      invariant { n = n0 > 100 \/ n0 <= n <= 101 } variant { 101 - n }
+      n <- f91_tco (n + 11)
+    done;
+    n - 10
+
   (** {2 non-recursive implementation using a while loop} *)
 
   use ref.Ref
   use int.Iter
 
-  let f91_nonrec (n0: int) ensures { result = spec n0 }
-  = let e = ref 1 in
-    let n = ref n0 in
-    while !e > 0 do
-      invariant { !e >= 0 /\ iter spec !e !n = spec n0 }
-      variant   { 101 - !n + 10 * !e, !e }
-      if !n > 100 then begin
-        n := !n - 10;
-        e := !e - 1
+  let f91_nonrec (n0: int): int
+    ensures { result = spec n0 }
+  = let ref e = 1 in
+    let ref n = n0 in
+    while e > 0 do
+      invariant { e >= 0 /\ iter spec e n = spec n0 }
+      variant   { 101 - n + 10 * e, e }
+      if n > 100 then begin
+        n <- n - 10;
+        e <- e - 1
       end else begin
-        n := !n + 11;
-        e := !e + 1
+        n <- n + 11;
+        e <- e + 1
       end
     done;
-    !n
+    n
 
   (** {2 irrelevance of control flow}
 
@@ -54,33 +66,33 @@ module McCarthy91
 
   exception Stop
 
-  let f91_pseudorec (n0:int) : int
+  let f91_pseudorec (n0: int) : int
     ensures { result = spec n0 }
-  = let e = ref 1 in
-    let n = ref n0 in
+  = let ref e = 1 in
+    let ref n = n0 in
     let bloc () : unit
-      requires { !e >= 0 }
-      ensures { !(old e) > 0 }
-      ensures { if !(old n) > 100 then !n = !(old n) - 10 /\ !e = !(old e) - 1
-        else !n = !(old n) + 11 /\ !e = !(old e) + 1 }
-      raises { Stop -> !e = !(old e) = 0 /\ !n = !(old n) }
-    = if not (!e > 0) then raise Stop;
-      if !n > 100 then begin
-        n := !n - 10;
-        e := !e - 1
+      requires { e >= 0 }
+      ensures { (old e) > 0 }
+      ensures { if (old n) > 100 then n = (old n) - 10 /\ e = (old e) - 1
+        else n = (old n) + 11 /\ e = (old e) + 1 }
+      raises { Stop -> e = (old e) = 0 /\ n = (old n) }
+    = if not (e > 0) then raise Stop;
+      if n > 100 then begin
+        n <- n - 10;
+        e <- e - 1
       end else begin
-        n := !n + 11;
-        e := !e + 1
+        n <- n + 11;
+        e <- e + 1
       end
     in
     let rec aux () : unit
-      requires { !e > 0 }
-      variant { 101 - !n }
-      ensures { !e = !(old e) - 1 /\ !n = spec !(old n) }
+      requires { e > 0 }
+      variant { 101 - n }
+      ensures { e = (old e) - 1 /\ n = spec (old n) }
       raises { Stop -> false }
-    = let u = !n in bloc (); if u <= 100 then (aux (); aux ()) in
+    = let u = n in bloc (); if u <= 100 then (aux (); aux ()) in
     try aux (); bloc (); absurd
-    with Stop -> !n end
+    with Stop -> n end
 
 
 end
@@ -119,31 +131,31 @@ we define the small-step semantics of this code by the following [step] function
 *)
 
 
-  val pc : ref int
-  val n : ref int
-  val e : ref int
-  val r : ref int
+  val ref pc : int
+  val ref n : int
+  val ref e : int
+  val ref r : int
 
   val step () : unit
-    requires { 0 <= !pc < 8 }
+    requires { 0 <= pc < 8 }
     writes   { pc, e, r }
-    ensures { old !pc = 0 -> !pc = 1 /\ !e = 1 /\ !r = old !r }
-    ensures { old !pc = 1 -> !pc = 2 /\ !e = old !e /\ !r = !n }
-    ensures { old !pc = 2 /\ old !r > 100 -> !pc = 3 /\ !e = old !e /\ !r = old !r }
-    ensures { old !pc = 2 /\ old !r <= 100 -> !pc = 6 /\ !e = old !e /\ !r = old !r }
-    ensures { old !pc = 3 -> !pc = 4 /\ !e = old !e /\ !r = old !r - 10 }
-    ensures { old !pc = 4 -> !pc = 5 /\ !e = old !e - 1 /\ !r = old !r }
-    ensures { old !pc = 5 /\ old !e = 0 -> !pc = 8 /\ !e = old !e /\ !r = old !r }
-    ensures { old !pc = 5 /\ old !e <> 0 -> !pc = 2 /\ !e = old !e /\ !r = old !r }
-    ensures { old !pc = 6 -> !pc = 7 /\ !e = old !e /\ !r = old !r + 11 }
-    ensures { old !pc = 7 -> !pc = 2 /\ !e = old !e + 1 /\ !r = old !r }
+    ensures { old pc = 0 -> pc = 1 /\ e = 1 /\ r = old r }
+    ensures { old pc = 1 -> pc = 2 /\ e = old e /\ r = n }
+    ensures { old pc = 2 /\ old r > 100 -> pc = 3 /\ e = old e /\ r = old r }
+    ensures { old pc = 2 /\ old r <= 100 -> pc = 6 /\ e = old e /\ r = old r }
+    ensures { old pc = 3 -> pc = 4 /\ e = old e /\ r = old r - 10 }
+    ensures { old pc = 4 -> pc = 5 /\ e = old e - 1 /\ r = old r }
+    ensures { old pc = 5 /\ old e = 0 -> pc = 8 /\ e = old e /\ r = old r }
+    ensures { old pc = 5 /\ old e <> 0 -> pc = 2 /\ e = old e /\ r = old r }
+    ensures { old pc = 6 -> pc = 7 /\ e = old e /\ r = old r + 11 }
+    ensures { old pc = 7 -> pc = 2 /\ e = old e + 1 /\ r = old r }
 
   let rec monitor () : unit
-    requires { !pc = 2 /\ !e > 0 }
-    variant  { 101 - !r }
-    ensures  { !pc = 5 /\ !r = spec(old !r) /\ !e = old !e - 1 }
+    requires { pc = 2 /\ e > 0 }
+    variant  { 101 - r }
+    ensures  { pc = 5 /\ r = spec(old r) /\ e = old e - 1 }
   = step (); (* execution of 'if r > 100' *)
-    if !pc = 3 then begin
+    if pc = 3 then begin
        step (); (* assignment r <- r - 10 *)
        step (); (* assignment e <- e - 1  *)
        end
@@ -156,8 +168,8 @@ we define the small-step semantics of this code by the following [step] function
        end
 
   let mccarthy ()
-    requires { !pc = 0 /\ !n >= 0 }
-    ensures { !pc = 8 /\ !r = spec !n }
+    requires { pc = 0 /\ n >= 0 }
+    ensures { pc = 8 /\ r = spec n }
   = step (); (* assignment e <- 1 *)
     step (); (* assignment r <- n *)
     monitor (); (* loop *)
@@ -183,29 +195,29 @@ we define the not-so-small-step semantics of this code by the following [next] f
 *)
 
   val next () : unit
-    requires { 0 <= !pc < 3 }
+    requires { 0 <= pc < 3 }
     writes   { pc, e, r }
-    ensures { old !pc = 0 -> !pc = 1 /\ !e = 1 /\ !r = !n }
-    ensures { old !pc = 1 /\ old !r > 100 ->
-              !pc = 2 /\ !r = old !r - 10 /\ !e = old !e - 1 }
-    ensures { old !pc = 1 /\ old !r <= 100 ->
-              !pc = 1 /\ !r = old !r + 11 /\ !e = old !e + 1 }
-    ensures { old !pc = 2 /\ old !e = 0 -> !pc = 3 /\ !r = old !r /\ !e = old !e }
-    ensures { old !pc = 2 /\ old !e <> 0 -> !pc = 1 /\ !r = old !r /\ !e = old !e }
+    ensures { old pc = 0 -> pc = 1 /\ e = 1 /\ r = n }
+    ensures { old pc = 1 /\ old r > 100 ->
+              pc = 2 /\ r = old r - 10 /\ e = old e - 1 }
+    ensures { old pc = 1 /\ old r <= 100 ->
+              pc = 1 /\ r = old r + 11 /\ e = old e + 1 }
+    ensures { old pc = 2 /\ old e = 0 -> pc = 3 /\ r = old r /\ e = old e }
+    ensures { old pc = 2 /\ old e <> 0 -> pc = 1 /\ r = old r /\ e = old e }
 
   (* [aux2] performs as may loop iterations as needed so as to reach program point 2
      from program point 1 *)
   let rec monitor2 () : unit
-    requires { !pc = 1 /\ !e > 0 }
-    variant  { 101 - !r }
-    ensures  { !pc = 2 /\ !r = spec(old !r) /\ !e = old !e - 1 }
+    requires { pc = 1 /\ e > 0 }
+    variant  { 101 - r }
+    ensures  { pc = 2 /\ r = spec(old r) /\ e = old e - 1 }
   = next ();
-    if !pc <> 2 then begin monitor2 (); next (); monitor2 () end
+    if pc <> 2 then begin monitor2 (); next (); monitor2 () end
 
 
   let mccarthy2 ()
-    requires { !pc = 0 /\ !n >= 0 }
-    ensures { !pc = 3 /\ !r = spec !n }
+    requires { pc = 0 /\ n >= 0 }
+    ensures { pc = 3 /\ r = spec n }
   = next (); (* assignments e <- 1; r <- n *)
     monitor2 (); (* loop *)
     next ()
@@ -233,49 +245,49 @@ module McCarthy91Mach
 
   let f91_nonrec (n0: int63) : int63
     ensures { result = spec n0 }
-  = let e = ref one in
-    let n = ref n0 in
-    while gt !e zero do
-      invariant { !e >= 0 /\ iter spec !e !n = spec n0 }
-      variant   { 101 - !n + 10 * !e, !e:int }
-      if !n > 100 then begin
-        n := !n - 10;
-        e := pred !e
+  = let ref e = one in
+    let ref n = n0 in
+    while gt e zero do
+      invariant { e >= 0 /\ iter spec e n = spec n0 }
+      variant   { 101 - n + 10 * e, e:int }
+      if n > 100 then begin
+        n <- n - 10;
+        e <- pred e
       end else begin
-        n := !n + 11;
-        e := succ !e
+        n <- n + 11;
+        e <- succ e
       end
     done;
-    !n
+    n
 
   exception Stop
 
   let f91_pseudorec (n0: int63) : int63
     ensures { result = spec n0 }
-  = let e = ref one in
-    let n = ref n0 in
+  = let ref e = one in
+    let ref n = n0 in
     let bloc () : unit
-      requires { !e >= 0 }
-      ensures { !(old e) > 0 }
-      ensures { if !(old n) > 100 then !n = !(old n) - 10 /\ !e = !(old e) - 1
-        else !n = !(old n) + 11 /\ !e = !(old e) + 1 }
-      raises { Stop -> !e = !(old e) = 0 /\ !n = !(old n) }
-    = if not (gt !e zero) then raise Stop;
-      if !n > 100 then begin
-        n := !n - 10;
-        e := pred !e
+      requires { e >= 0 }
+      ensures { (old e) > 0 }
+      ensures { if (old n) > 100 then n = (old n) - 10 /\ e = (old e) - 1
+        else n = (old n) + 11 /\ e = (old e) + 1 }
+      raises { Stop -> e = (old e) = 0 /\ n = (old n) }
+    = if not (gt e zero) then raise Stop;
+      if n > 100 then begin
+        n := n - 10;
+        e := pred e
       end else begin
-        n := !n + 11;
-        e := succ !e
+        n := n + 11;
+        e := succ e
       end
     in
     let rec aux () : unit
-      requires { !e > 0 }
-      variant { 101 - !n }
-      ensures { !e = !(old e) - 1 /\ !n = spec !(old n) }
+      requires { e > 0 }
+      variant { 101 - n }
+      ensures { e = (old e) - 1 /\ n = spec (old n) }
       raises { Stop -> false }
-    = let u = !n in bloc (); if u <= 100 then (aux (); aux ()) in
+    = let u = n in bloc (); if u <= 100 then (aux (); aux ()) in
     try aux (); bloc (); absurd
-    with Stop -> !n end
+    with Stop -> n end
 
 end

examples/mccarthy/why3session.xml

@@ -4,11 +4,39 @@
 <why3session shape_version="5">
 <prover id="1" name="CVC4" version="1.6" timelimit="1" steplimit="0" memlimit="1000"/>
 <prover id="3" name="Alt-Ergo" version="2.0.0" timelimit="10" steplimit="0" memlimit="1000"/>
-<file name="../mccarthy.mlw" proved="true">
+<file proved="true">
+<path name=".."/>
+<path name="mccarthy.mlw"/>
 <theory name="McCarthy91" proved="true">
  <goal name="VC f91" expl="VC for f91" proved="true">
  <proof prover="3"><result status="valid" time="0.00" steps="31"/></proof>
  </goal>
+ <goal name="VC f91_tco" expl="VC for f91_tco" proved="true">
+ <transf name="split_vc" proved="true" >
+  <goal name="VC f91_tco.0" expl="loop invariant init" proved="true">
+  <proof prover="1"><result status="valid" time="0.04"/></proof>
+  </goal>
+  <goal name="VC f91_tco.1" expl="variant decrease" proved="true">
+  <transf name="split_vc" proved="true" >
+   <goal name="VC f91_tco.1.0" expl="variant decrease" proved="true">
+   <proof prover="1"><result status="valid" time="0.04"/></proof>
+   </goal>
+   <goal name="VC f91_tco.1.1" expl="variant decrease" proved="true">
+   <proof prover="1"><result status="valid" time="0.02"/></proof>
+   </goal>
+  </transf>
+  </goal>
+  <goal name="VC f91_tco.2" expl="loop variant decrease" proved="true">
+  <proof prover="1"><result status="valid" time="0.04"/></proof>
+  </goal>
+  <goal name="VC f91_tco.3" expl="loop invariant preservation" proved="true">
+  <proof prover="1"><result status="valid" time="0.02"/></proof>
+  </goal>
+  <goal name="VC f91_tco.4" expl="postcondition" proved="true">
+  <proof prover="1"><result status="valid" time="0.02"/></proof>
+  </goal>
+ </transf>
+ </goal>
  <goal name="VC f91_nonrec" expl="VC for f91_nonrec" proved="true">
  <proof prover="3"><result status="valid" time="0.29" steps="523"/></proof>
  </goal>