src/HOL/ex/Fib.ML
 author wenzelm Mon, 03 Nov 1997 12:13:18 +0100 changeset 4089 96fba19bcbe2 parent 3919 c036caebfc75 child 4379 7049ca8f912e permissions -rw-r--r--
isatool fixclasimp;
```
(*  Title:      HOL/ex/Fib
ID:         \$Id\$
Author:     Lawrence C Paulson

Fibonacci numbers: proofs of laws taken from

R. L. Graham, D. E. Knuth, O. Patashnik.
Concrete Mathematics.
*)

(** The difficulty in these proofs is to ensure that the induction hypotheses
are applied before the definition of "fib".  Towards this end, the
"fib" equations are not added to the simpset and are applied very
selectively at first.
**)

bind_thm ("fib_Suc_Suc", hd(rev fib.rules));

(*Concrete Mathematics, page 280*)
goal thy "fib (Suc (n + k)) = fib(Suc k) * fib(Suc n) + fib k * fib n";
by (res_inst_tac [("u","n")] fib.induct 1);
(*Simplify the LHS just enough to apply the induction hypotheses*)
by (asm_full_simp_tac
by (ALLGOALS
(fib.rules @ add_ac @ mult_ac @

goal thy "fib (Suc n) ~= 0";
by (res_inst_tac [("u","n")] fib.induct 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps fib.rules)));
qed "fib_Suc_neq_0";

(*Concrete Mathematics, page 278: Cassini's identity*)
goal thy "fib (Suc (Suc n)) * fib n = \
\              (if n mod 2 = 0 then pred(fib(Suc n) * fib(Suc n)) \
\                              else Suc (fib(Suc n) * fib(Suc n)))";
by (res_inst_tac [("u","n")] fib.induct 1);
by (res_inst_tac [("P", "%z. ?ff(x) * z = ?kk(x)")] (fib_Suc_Suc RS ssubst) 3);
by (stac (read_instantiate [("x", "Suc(Suc ?n)")] fib_Suc_Suc) 3);
by (stac (read_instantiate [("x", "Suc ?n")] fib_Suc_Suc) 3);
by (ALLGOALS  (*using fib.rules here results in a longer proof!*)