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(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA LRI-CNRS ENS-CNRS *)
(* Rocquencourt Orsay Lyon *)
(* *)
(* Coq V6.3 *)
(* July 1st 1999 *)
(* *)
(****************************************************************************)
(* Certification of Imperative Programs *)
(* Jean-Christophe Filliâtre *)
(****************************************************************************)
(* Handbook.v *)
(****************************************************************************)
(* This file contains proofs of programs taken from the
* "Handbook of Theoretical Computer Science", volume B,
* chapter "Methods and Logics for Proving Programs", by P. Cousot,
* pp 841--993, Edited by J. van Leeuwen (c) Elsevier Science Publishers B.V.
* 1990.
*
* Programs are refered to by numbers and pages.
*)
Require Programs.
Require Sumbool.
Require Omega.
Require Zcomplements.
Require Zpower.
(****************************************************************************)
(* program (2) page 853 to compute x^y (annotated version is (25) page 860) *)
(* en attendant... *)
Parameter Zdiv2 : Z->Z.
Parameter Zeven_odd_dec : (x:Z){`x=2*(Zdiv2 x)`}+{`x=2*(Zdiv2 x)+1`}.
Definition Zodd_dec := [z:Z](sumbool_not ? ? (Zeven_odd_dec z)).
Definition Zodd_bool := [z:Z](bool_of_sumbool ? ? (Zodd_dec z)).
Axiom axiom1 : (x,y:Z) `y>0` -> `x*(Zpower x (Zpred y)) = (Zpower x y)`.
Axiom axiom2 : (x:Z)`x>0` -> `(Zdiv2 x)<x`.
Axiom axiom3 : (x,y:Z) `y>=0` -> `(Zpower (x*x) (Zdiv2 y)) = (Zpower x y)`.
Global Variable X : Z ref.
Global Variable Y : Z ref.
Global Variable Z : Z ref.
Correctness pgm25
{ `Y >= 0` }
begin
Z := 1;
while !Y <> 0 do
{ invariant `Y >= 0` /\ `Z * (Zpower X Y) = (Zpower X@0 Y@0)`
variant Y }
if (Zodd_bool !Y) then begin
Y := (Zpred !Y);
Z := (Zmult !Z !X)
end else begin
Y := (Zdiv2 !Y);
X := (Zmult !X !X)
end
done
end
{ Z = (Zpower X@ Y@) }.
Proof.
Split.
Unfold Zpred; Unfold Zwf; Omega.
Split.
Unfold Zpred; Omega.
Decompose [and] Pre2.
Rewrite <- H1.
Replace `Z1*X0*(Zpower X0 (Zpred Y0))` with `Z1*(X0*(Zpower X0 (Zpred Y0)))`.
Apply f_equal with f := (Zmult Z1).
Apply axiom1.
Omega.
Auto.
Symmetry.
Apply Zmult_assoc_r.
Split.
Unfold Zwf.
Repeat (Apply conj).
Omega.
Omega.
Apply axiom2. Omega.
Split.
Omega.
Decompose [and] Pre2.
Rewrite <- H1.
Apply f_equal with f:=(Zmult Z1).
Apply axiom3. Omega.
Omega.
Decompose [and] Post6.
Rewrite <- H2.
Rewrite H1.
Simpl.
Omega.
Save.
(****************************************************************************)
(* program (178) page 934 to compute the factorial using global variables
* annotated version is (185) page 939
*)
Parameter Zfact : Z -> Z.
Axiom axiom4 : `(Zfact 0) = 1`.
Axiom axiom5 : (x:Z) `x>0` -> `(Zfact (x-1))*x=(Zfact x)`.
Correctness pgm178
let rec F (u:unit) : unit { variant X } =
{ `X>=0` }
(if !X = 0 then
Y := 1
else begin
label L;
X := (Zpred !X);
(F tt);
X := (Zs !X);
Y := (Zmult !Y !X)
end)
{ `X=X@` /\ `Y=(Zfact X@)` }.
Proof.
Rewrite Test1. Rewrite axiom4. Auto.
Unfold Zwf. Unfold Zpred. Omega.
Unfold Zpred. Omega.
Unfold Zs. Unfold Zpred in Post3. Split.
Omega.
Decompose [and] Post3.
Rewrite H0.
Replace `X0+(-1)+1` with X0.
Rewrite H1.
Replace `X0+(-1)` with `X0-1`.
Apply axiom5.
Omega.
Omega.
Omega.
Save.
(****************************************************************************)
(* program (186) page 939 "showing the usefulness of auxiliary variables" ! *)
Global Variable N : Z ref.
Global Variable S : Z ref.
Correctness pgm186
let rec F (u:unit) : unit { variant N } =
{ `N>=0` }
(if !N > 0 then begin
label L;
N := (Zpred !N);
(F tt);
S := (Zs !S);
(F tt);
N := (Zs !N)
end)
{ `N=N@` /\ `S=S@+(Zpower 2 N@)-1` }.
Proof.
Unfold Zwf. Unfold Zpred. Omega.
Unfold Zpred. Omega.
Decompose [and] Post5. Rewrite H0. Unfold Zwf. Unfold Zpred. Omega.
Decompose [and] Post5. Rewrite H0. Unfold Zpred. Omega.
Split.
Unfold Zpred in Post5. Omega.
Decompose [and] Post4. Rewrite H1.
Decompose [and] Post5. Rewrite H3. Rewrite H2.
Replace `(Zpower 2 N0)` with `2*(Zpower 2 (Zpred N0))`. Omega.
Symmetry.
Replace `(Zpower 2 N0)` with `(Zpower 2 (1+(Zpred N0)))`.
Replace `2*(Zpower 2 (Zpred N0))` with `(Zpower 2 1)*(Zpower 2 (Zpred N0))`.
Apply Zpower_exp.
Omega.
Unfold Zpred. Omega.
Auto.
Replace `(1+(Zpred N0))` with N0; [ Auto | Unfold Zpred; Omega ].
Split.
Auto.
Replace N0 with `0`; Simpl; Omega.
Save.
(****************************************************************************)
(* program (196) page 944 (recursive factorial procedure with value-result
* parameters)
*)
Correctness pgm196
let rec F (U:Z) (V:Z ref) : unit { variant U } =
{ `U >= 0` }
(if U = 0 then
V := 1
else begin
(F (Zpred U) V);
V := (Zmult !V U)
end)
{ `V = (Zfact U)` }.
Proof.
Symmetry. Rewrite Test1. Apply axiom4.
Unfold Zwf. Unfold Zpred. Omega.
Unfold Zpred. Omega.
Rewrite Post3.
Unfold Zpred. Replace `U0+(-1)` with `U0-1`. Apply axiom5.
Omega.
Omega.
Save.
(****************************************************************************)
(* program (197) page 945 (L_4 subset of Pascal) *)
(*
procedure P(X:Z; procedure Q(Z:Z));
procedure L(X:Z); begin Q(X-1) end;
begin if X>0 then P(X-1,L) else Q(X) end;
procedure M(N:Z);
procedure R(X:Z); begin writeln(X) (* => RES := !X *) end;
begin P(N,R) end.
*)
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