theory FibonacciTest use import int.Fibonacci lemma isfib_2_1 : fib 2 = 1 lemma isfib_6_8 : fib 6 = 8 lemma not_isfib_2_2 : fib 2 <> 2 end module FibonacciLinear use import int.Fibonacci use import int.Int use import ref.Ref let fib (n:int) : int requires { n >= 0 } ensures { fib n = result} = let y = ref 0 in let x = ref 1 in for i = 0 to n - 1 do invariant { 0 <= i <= n /\ fib (i+1) = !x /\ fib i = !y } let aux = !y in y := !x; x := !x + aux done; !y end module FibRecGhost "recursive version, using ghost code" use import int.Fibonacci use import int.Int let rec fib_aux (ghost n: int) (a b k: int) : int requires { k >= 0 } requires { 0 <= n && a = fib n && b = fib (n+1) } variant { k } ensures { result = fib (n+k) } = if k = 0 then a else fib_aux (n+1) b (a+b) (k-1) let fib (n: int) : int requires { 0 <= n } ensures { result = fib n } = fib_aux 0 0 1 n let test42 () = fib 42 exception BenchFailure let bench () raises { BenchFailure -> true } = if test42 () <> 267914296 then raise BenchFailure end module FibRecNoGhost "recursive version, without ghost code" use import int.Fibonacci use import int.Int let rec fib_aux (a b k: int) : int requires { k >= 0 } requires { exists n: int. 0 <= n && a = fib n && b = fib (n+1) } variant { k } ensures { forall n: int. 0 <= n && a = fib n && b = fib (n+1) -> result = fib (n+k) } = if k = 0 then a else fib_aux b (a+b) (k-1) let fib (n: int) : int requires { 0 <= n } ensures { result = fib n } = fib_aux 0 1 n end module SmallestFibAbove use import int.Fibonacci use import int.Int use import int.MinMax use import ref.Ref let smallest_fib_above (x: int) : int requires { 0 <= x } ensures { exists k: int. 0 <= k /\ fib k <= x < fib (k+1) = result } = let a = ref 0 in let b = ref 1 in while !b <= x do invariant { exists k: int. 0 <= k /\ !a = fib k <= x /\ !b = fib (k+1) } invariant { 0 <= !a /\ 1 <= !b } variant { 2*x - (!a + !b) } let f = !a + !b in a := !b; b := f done; !b end (** Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. Cf https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem *) module Zeckendorf use import int.Fibonacci use import int.Int use import int.MinMax use import ref.Ref use import list.List use import SmallestFibAbove function sum (l: list int) : int = match l with | Nil -> 0 | Cons k r -> fib k + sum r end (* sorted in increasing order, above min, and non consecutive *) predicate wf (min: int) (l: list int) = match l with | Nil -> true | Cons k r -> min <= k /\ wf (k + 2) r end let rec lemma fib_nonneg (n: int) : unit requires { 0 <= n } ensures { 0 <= fib n } variant { n } = if n > 1 then begin fib_nonneg (n-2); fib_nonneg (n-1) end let rec lemma fib_increasing (k1 k2: int) : unit requires { 0 <= k1 <= k2 } ensures { fib k1 <= fib k2 } variant { k2 - k1 } = if k1 < k2 then fib_increasing (k1+1) k2 let greatest_fib (x: int) : (int, int) requires { 0 < x } ensures { let (k, fk) = result in 2 <= k /\ 1 <= fk = fib k <= x < fib (k + 1) } = let a = ref 1 in let b = ref 1 in let k = ref 1 in while !b <= x do invariant { 1 <= !k /\ !a = fib !k <= x /\ !b = fib (!k + 1) } invariant { 1 <= !a /\ 1 <= !b } variant { 2*x - (!a + !b) } let f = !a + !b in a := !b; b := f; k := !k + 1 done; (!k, !a) let zeckendorf (n: int) : list int requires { 0 <= n } ensures { wf 2 result } ensures { n = sum result } = let x = ref n in let l = ref Nil in while !x > 0 do invariant { 0 <= !x <= n } invariant { wf 2 !l } invariant { !x + sum !l = n } invariant { match !l with Nil -> true | Cons k _ -> !x < fib (k-1) end } variant { !x } let (k, fk) = greatest_fib !x in x := !x - fk; l := Cons k !l done; !l (* a more efficient, linear implementation *) let zeckendorf_fast (n: int) : list int requires { 0 <= n } ensures { wf 2 result } ensures { n = sum result } = if n = 0 then Nil else let a = ref 1 in let b = ref 1 in let k = ref 1 in while !b <= n do invariant { 1 <= !k /\ !a = fib !k <= n /\ !b = fib (!k + 1) } invariant { 1 <= !a /\ 1 <= !b } variant { 2*n - (!a + !b) } let f = !a + !b in a := !b; b := f; k := !k + 1 done; assert { 2 <= !k /\ 1 <= !a = fib !k <= n < fib (!k + 1) = !b }; let l = ref (Cons !k Nil) in let x = ref (n - !a) in while !x > 0 do invariant { 1 <= !k /\ !a = fib !k <= n /\ !x < !b = fib (!k + 1) } invariant { 1 <= !a /\ 1 <= !b } invariant { 0 <= !x <= n } invariant { wf 2 !l } invariant { !x + sum !l = n } invariant { match !l with Nil -> true | Cons k _ -> !x < fib (k-1) end } variant { !k } if !a <= !x then begin x := !x - !a; l := Cons !k !l end; k := !k - 1; let tmp = !b - !a in b := !a; a := tmp done; !l (* unicity proof *) function snoc (l:list int) (x:int) : list int = match l with | Nil -> Cons x Nil | Cons y q -> Cons y (snoc q x) end let rec lemma zeckendorf_unique (l1 l2:list int) : unit requires { wf 2 l1 /\ wf 2 l2 } requires { sum l1 = sum l2 } ensures { l1 = l2 } variant { sum l1 } = let rec decomp (k acc:int) (lc lb:list int) : (int,list int) requires { wf k lc } requires { k >= 2 /\ lc <> Nil } requires { 0 <= acc = sum lb - sum lc < fib (k-1) } returns { (x,p) -> fib x <= sum lb = acc + fib x + sum p < fib (x+1) } returns { (x,p) -> wf k p /\ x >= k /\ lc = snoc p x } variant { lc } = match lc with | Nil -> absurd | Cons x Nil -> (x,Nil) | Cons x q -> let (y,p) = decomp (x+2) (acc+fib x) q lb in (y,Cons x p) end in match l1 , l2 with | Nil , Nil -> () | Nil , l | l , Nil -> let _ = decomp 2 0 l l in absurd | _ , _ -> let (a1,q1) = decomp 2 0 l1 l1 in let (a2,q2) = decomp 2 0 l2 l2 in zeckendorf_unique q1 q2 end end theory Mat22 "2x2 integer matrices" use import int.Int type t = { a11: int; a12: int; a21: int; a22: int } constant id : t = { a11 = 1; a12 = 0; a21 = 0; a22 = 1 } function mult (x: t) (y: t) : t = { a11 = x.a11 * y.a11 + x.a12 * y.a21; a12 = x.a11 * y.a12 + x.a12 * y.a22; a21 = x.a21 * y.a11 + x.a22 * y.a21; a22 = x.a21 * y.a12 + x.a22 * y.a22; } (* holds, but not useful *) (* clone algebra.Assoc with type t = t, function op = mult, lemma Assoc *) clone export int.Exponentiation with type t = t, function one = id, function (*) = mult end module FibonacciLogarithmic use import int.Int use import int.Fibonacci use import int.EuclideanDivision use import Mat22 constant m1110 : t = { a11 = 1; a12 = 1; a21 = 1; a22 = 0 } (* computes ((1 1) (1 0))^n in O(log(n)) time since it is a matrix of the shape ((a+b b) (b a)), we only return the pair (a, b) *) let rec logfib n variant { n } requires { n >= 0 } ensures { let a, b = result in power m1110 n = { a11 = a+b; a12 = b; a21 = b; a22 = a } } = if n = 0 then (1, 0) else begin let a, b = logfib (div n 2) in let c = a + b in if mod n 2 = 0 then (a*a + b*b, b*(a + c)) else (b*(a + c), c*c + b*b) end (* by induction, we easily prove that (1 1)^n = (F(n+1) F(n) ) (1 0) (F(n) F(n-1)) thus, we can compute F(n) in O(log(n)) using funtion logfib above *) lemma fib_m : forall n: int. n >= 0 -> let p = power m1110 n in fib (n+1) = p.a11 /\ fib n = p.a21 let fibo n requires { n >= 0 } ensures { result = fib n } = let _, b = logfib n in b let test0 () = fibo 0 let test1 () = fibo 1 let test7 () = fibo 7 let test42 () = fibo 42 let test2014 () = fibo 2014 exception BenchFailure let bench () raises { BenchFailure -> true } = if test42 () <> 267914296 then raise BenchFailure; if test2014 () <> 3561413997540486142674781564382874188700994538849211456995042891654110985470076818421080236961243875711537543388676277339875963824466334432403730750376906026741819889036464401788232213002522934897299928844192803507157647764542466327613134605502785287441134627457615461304177503249289874066244145666889138852687147544158443155204157950294129177785119464446668374163746700969372438526182906768143740891051274219441912520127 then raise BenchFailure end