Files for testing theorem dependency features.

6 files changed, 361 insertions, 0 deletions

diff --git a/etc/isa/depends/Fib.ML b/etc/isa/depends/Fib.ML
new file mode 100644
index 00000000..eba2f0e0
--- /dev/null
+++ b/etc/isa/depends/Fib.ML
@@ -0,0 +1,106 @@
+(*  Title:      HOL/ex/Fib
+    ID:         $Id$
+    Author:     Lawrence C Paulson
+    Copyright   1997  University of Cambridge
+
+Fibonacci numbers: proofs of laws taken from
+
+  R. L. Graham, D. E. Knuth, O. Patashnik.
+  Concrete Mathematics.
+  (Addison-Wesley, 1989)
+*)
+
+
+(** 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.
+**)
+
+Delsimps fib.Suc_Suc;
+
+val [fib_Suc_Suc] = fib.Suc_Suc;
+val fib_Suc3 = read_instantiate [("x", "(Suc ?n)")] fib_Suc_Suc;
+
+(*Concrete Mathematics, page 280*)
+Goal "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
+    (simpset() addsimps [inst "x" "Suc(?m+?n)" fib_Suc_Suc]) 3);
+by (ALLGOALS 
+    (asm_simp_tac (simpset() addsimps 
+		   ([fib_Suc_Suc, add_mult_distrib, add_mult_distrib2]))));
+qed "fib_add";
+
+
+Goal "fib (Suc n) ~= 0";
+by (res_inst_tac [("u","n")] fib.induct 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [fib_Suc_Suc])));
+qed "fib_Suc_neq_0";
+
+(* Also add  0 < fib (Suc n) *)
+Addsimps [fib_Suc_neq_0, [neq0_conv, fib_Suc_neq_0] MRS iffD1];
+
+Goal "0<n ==> 0 < fib n";
+by (rtac (not0_implies_Suc RS exE) 1);
+by Auto_tac;
+qed "fib_gr_0";
+
+(*Concrete Mathematics, page 278: Cassini's identity.
+  It is much easier to prove using integers!*)
+Goal "int (fib (Suc (Suc n)) * fib n) = \
+\     (if n mod 2 = 0 then int (fib(Suc n) * fib(Suc n)) - #1 \
+\                     else int (fib(Suc n) * fib(Suc n)) + #1)";
+by (res_inst_tac [("u","n")] fib.induct 1);
+by (simp_tac (simpset() addsimps [fib_Suc_Suc, mod_Suc]) 2);
+by (simp_tac (simpset() addsimps [fib_Suc_Suc]) 1);
+by (asm_full_simp_tac
+     (simpset() addsimps [fib_Suc_Suc, add_mult_distrib, add_mult_distrib2, 
+			  mod_Suc, zmult_int RS sym] @ zmult_ac) 1);
+qed "fib_Cassini";
+
+
+
+(** Towards Law 6.111 of Concrete Mathematics **)
+
+Goal "gcd(fib n, fib (Suc n)) = 1";
+by (res_inst_tac [("u","n")] fib.induct 1);
+by (asm_simp_tac (simpset() addsimps [fib_Suc3, gcd_commute, gcd_add2]) 3);
+by (ALLGOALS (simp_tac (simpset() addsimps [fib_Suc_Suc])));
+qed "gcd_fib_Suc_eq_1"; 
+
+val gcd_fib_commute = 
+    read_instantiate_sg (sign_of thy) [("m", "fib m")] gcd_commute;
+
+Goal "gcd(fib m, fib (n+m)) = gcd(fib m, fib n)";
+by (simp_tac (simpset() addsimps [gcd_fib_commute]) 1);
+by (case_tac "m=0" 1);
+by (Asm_simp_tac 1);
+by (clarify_tac (claset() addSDs [not0_implies_Suc]) 1);
+by (simp_tac (simpset() addsimps [fib_add]) 1);
+by (asm_simp_tac (simpset() addsimps [add_commute, gcd_non_0]) 1);
+by (asm_simp_tac (simpset() addsimps [gcd_non_0 RS sym]) 1);
+by (asm_simp_tac (simpset() addsimps [gcd_fib_Suc_eq_1, gcd_mult_cancel]) 1);
+qed "gcd_fib_add";
+
+Goal "m <= n ==> gcd(fib m, fib (n-m)) = gcd(fib m, fib n)";
+by (rtac (gcd_fib_add RS sym RS trans) 1);
+by (Asm_simp_tac 1);
+qed "gcd_fib_diff";
+
+Goal "0<m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)";
+by (res_inst_tac [("n","n")] less_induct 1);
+by (stac mod_if 1);
+by (Asm_simp_tac 1);
+by (asm_simp_tac (simpset() addsimps [gcd_fib_diff, mod_geq, 
+				      not_less_iff_le, diff_less]) 1);
+qed "gcd_fib_mod";
+
+(*Law 6.111*)
+Goal "fib(gcd(m,n)) = gcd(fib m, fib n)";
+by (res_inst_tac [("m","m"),("n","n")] gcd_induct 1);
+by (Asm_simp_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [gcd_non_0]) 1);
+by (asm_full_simp_tac (simpset() addsimps [gcd_commute, gcd_fib_mod]) 1);
+qed "fib_gcd";
diff --git a/etc/isa/depends/Fib.thy b/etc/isa/depends/Fib.thy
new file mode 100644
index 00000000..9272ed8c
--- /dev/null
+++ b/etc/isa/depends/Fib.thy
@@ -0,0 +1,17 @@
+(*  Title:      ex/Fib
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1997  University of Cambridge
+
+The Fibonacci function.  Demonstrates the use of recdef.
+*)
+
+Fib = Usedepends + Divides + Primes +
+
+consts fib  :: "nat => nat"
+recdef fib "less_than"
+  zero    "fib 0 = 0"
+  one     "fib 1 = 1"
+  Suc_Suc "fib (Suc (Suc x)) = fib x + fib (Suc x)"
+
+end
diff --git a/etc/isa/depends/Primes.ML b/etc/isa/depends/Primes.ML
new file mode 100644
index 00000000..102419da
--- /dev/null
+++ b/etc/isa/depends/Primes.ML
@@ -0,0 +1,197 @@
+(*  Title:      HOL/ex/Primes.ML
+    ID:         $Id$
+    Author:     Christophe Tabacznyj and Lawrence C Paulson
+    Copyright   1996  University of Cambridge
+
+The "divides" relation, the greatest common divisor and Euclid's algorithm
+
+See H. Davenport, "The Higher Arithmetic".  6th edition.  (CUP, 1992)
+*)
+
+eta_contract:=false;
+
+(************************************************)
+(** Greatest Common Divisor                    **)
+(************************************************)
+
+(*** Euclid's Algorithm ***)
+
+
+val [gcd_eq] = gcd.simps;
+
+
+val prems = goal thy
+     "[| !!m. P m 0;     \
+\        !!m n. [| 0<n;  P n (m mod n) |] ==> P m n  \
+\     |] ==> P (m::nat) (n::nat)";
+by (res_inst_tac [("u","m"),("v","n")] gcd.induct 1);
+by (case_tac "n=0" 1);
+by (asm_simp_tac (simpset() addsimps prems) 1);
+by Safe_tac;
+by (asm_simp_tac (simpset() addsimps prems) 1);
+qed "gcd_induct";
+
+
+Goal "gcd(m,0) = m";
+by (Simp_tac 1);
+qed "gcd_0";
+Addsimps [gcd_0];
+
+Goal "0<n ==> gcd(m,n) = gcd (n, m mod n)";
+by (Asm_simp_tac 1);
+qed "gcd_non_0";
+
+Delsimps gcd.simps;
+
+Goal "gcd(m,1) = 1";
+by (simp_tac (simpset() addsimps [gcd_non_0]) 1);
+qed "gcd_1";
+Addsimps [gcd_1];
+
+(*gcd(m,n) divides m and n.  The conjunctions don't seem provable separately*)
+Goal "(gcd(m,n) dvd m) & (gcd(m,n) dvd n)";
+by (res_inst_tac [("m","m"),("n","n")] gcd_induct 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [gcd_non_0])));
+by (blast_tac (claset() addDs [dvd_mod_imp_dvd]) 1);
+qed "gcd_dvd_both";
+
+bind_thm ("gcd_dvd1", gcd_dvd_both RS conjunct1);
+bind_thm ("gcd_dvd2", gcd_dvd_both RS conjunct2);
+
+
+(*Maximality: for all m,n,f naturals, 
+                if f divides m and f divides n then f divides gcd(m,n)*)
+Goal "(f dvd m) --> (f dvd n) --> f dvd gcd(m,n)";
+by (res_inst_tac [("m","m"),("n","n")] gcd_induct 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps[gcd_non_0, dvd_mod])));
+qed_spec_mp "gcd_greatest";
+
+(*Function gcd yields the Greatest Common Divisor*)
+Goalw [is_gcd_def] "is_gcd (gcd(m,n)) m n";
+by (asm_simp_tac (simpset() addsimps [gcd_greatest, gcd_dvd_both]) 1);
+qed "is_gcd";
+
+(*uniqueness of GCDs*)
+Goalw [is_gcd_def] "[| is_gcd m a b; is_gcd n a b |] ==> m=n";
+by (blast_tac (claset() addIs [dvd_anti_sym]) 1);
+qed "is_gcd_unique";
+
+(*USED??*)
+Goalw [is_gcd_def]
+    "[| is_gcd m a b; k dvd a; k dvd b |] ==> k dvd m";
+by (Blast_tac 1);
+qed "is_gcd_dvd";
+
+(** Commutativity **)
+
+Goalw [is_gcd_def] "is_gcd k m n = is_gcd k n m";
+by (Blast_tac 1);
+qed "is_gcd_commute";
+
+Goal "gcd(m,n) = gcd(n,m)";
+by (rtac is_gcd_unique 1);
+by (rtac is_gcd 2);
+by (asm_simp_tac (simpset() addsimps [is_gcd, is_gcd_commute]) 1);
+qed "gcd_commute";
+
+Goal "gcd(gcd(k,m),n) = gcd(k,gcd(m,n))";
+by (rtac is_gcd_unique 1);
+by (rtac is_gcd 2);
+by (rewtac is_gcd_def);
+by (blast_tac (claset() addSIs [gcd_dvd1, gcd_dvd2]
+   	                addIs  [gcd_greatest, dvd_trans]) 1);
+qed "gcd_assoc";
+
+Goal "gcd(0,m) = m";
+by (stac gcd_commute 1);
+by (rtac gcd_0 1);
+qed "gcd_0_left";
+
+Goal "gcd(1,m) = 1";
+by (stac gcd_commute 1);
+by (rtac gcd_1 1);
+qed "gcd_1_left";
+Addsimps [gcd_0_left, gcd_1_left];
+
+
+(** Multiplication laws **)
+
+(*Davenport, page 27*)
+Goal "k * gcd(m,n) = gcd(k*m, k*n)";
+by (res_inst_tac [("m","m"),("n","n")] gcd_induct 1);
+by (Asm_full_simp_tac 1);
+by (case_tac "k=0" 1);
+ by (Asm_full_simp_tac 1);
+by (asm_full_simp_tac
+    (simpset() addsimps [mod_geq, gcd_non_0, mod_mult_distrib2]) 1);
+qed "gcd_mult_distrib2";
+
+Goal "gcd(m,m) = m";
+by (cut_inst_tac [("k","m"),("m","1"),("n","1")] gcd_mult_distrib2 1);
+by (Asm_full_simp_tac 1);
+qed "gcd_self";
+Addsimps [gcd_self];
+
+Goal "gcd(k, k*n) = k";
+by (cut_inst_tac [("k","k"),("m","1"),("n","n")] gcd_mult_distrib2 1);
+by (Asm_full_simp_tac 1);
+qed "gcd_mult";
+Addsimps [gcd_mult];
+
+Goal "[| gcd(k,n)=1; k dvd (m*n) |] ==> k dvd m";
+by (subgoal_tac "m = gcd(m*k, m*n)" 1);
+by (etac ssubst 1 THEN rtac gcd_greatest 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [gcd_mult_distrib2 RS sym])));
+qed "relprime_dvd_mult";
+
+Goalw [prime_def] "[| p: prime;  ~ p dvd n |] ==> gcd (p, n) = 1";
+by (cut_inst_tac [("m","p"),("n","n")] gcd_dvd_both 1);
+by Auto_tac;
+qed "prime_imp_relprime";
+
+(*This theorem leads immediately to a proof of the uniqueness of factorization.
+  If p divides a product of primes then it is one of those primes.*)
+Goal "[| p: prime; p dvd (m*n) |] ==> p dvd m | p dvd n";
+by (blast_tac (claset() addIs [relprime_dvd_mult, prime_imp_relprime]) 1);
+qed "prime_dvd_mult";
+
+
+(** Addition laws **)
+
+Goal "gcd(m, m+n) = gcd(m,n)";
+by (res_inst_tac [("n1", "m+n")] (gcd_commute RS ssubst) 1);
+by (rtac (gcd_eq RS trans) 1);
+by Auto_tac;
+by (asm_simp_tac (simpset() addsimps [mod_add_self1]) 1);
+by (stac gcd_commute 1);
+by (stac gcd_non_0 1);
+by Safe_tac;
+qed "gcd_add";
+
+Goal "gcd(m, n+m) = gcd(m,n)";
+by (asm_simp_tac (simpset() addsimps [add_commute, gcd_add]) 1);
+qed "gcd_add2";
+
+Goal "gcd(m, k*m+n) = gcd(m,n)";
+by (induct_tac "k" 1);
+by (asm_simp_tac (simpset() addsimps [gcd_add, add_assoc]) 2); 
+by (Simp_tac 1);
+qed "gcd_add_mult";
+
+
+(** More multiplication laws **)
+
+Goal "gcd(m,n) dvd gcd(k*m, n)";
+by (blast_tac (claset() addIs [gcd_greatest, dvd_trans, 
+                               gcd_dvd1, gcd_dvd2]) 1);
+qed "gcd_dvd_gcd_mult";
+
+Goal "gcd(k,n) = 1 ==> gcd(k*m, n) = gcd(m,n)";
+by (rtac dvd_anti_sym 1);
+by (rtac gcd_dvd_gcd_mult 2);
+by (rtac ([relprime_dvd_mult, gcd_dvd2] MRS gcd_greatest) 1);
+by (stac mult_commute 2);
+by (rtac gcd_dvd1 2);
+by (stac gcd_commute 1);
+by (asm_simp_tac (simpset() addsimps [gcd_assoc RS sym]) 1);
+qed "gcd_mult_cancel";
diff --git a/etc/isa/depends/Primes.thy b/etc/isa/depends/Primes.thy
new file mode 100644
index 00000000..fac39463
--- /dev/null
+++ b/etc/isa/depends/Primes.thy
@@ -0,0 +1,33 @@
+(*  Title:      HOL/ex/Primes.thy
+    ID:         $Id$
+    Author:     Christophe Tabacznyj and Lawrence C Paulson
+    Copyright   1996  University of Cambridge
+
+The Greatest Common Divisor and Euclid's algorithm
+
+The "simpset" clause in the recdef declaration used to be necessary when the
+two lemmas where not part of the default simpset.
+*)
+
+Primes = Main +
+
+
+consts
+  is_gcd  :: [nat,nat,nat]=>bool          (*gcd as a relation*)
+  gcd     :: "nat*nat=>nat"               (*Euclid's algorithm *)
+  coprime :: [nat,nat]=>bool
+  prime   :: nat set
+  
+recdef gcd "measure ((%(m,n).n) ::nat*nat=>nat)"
+(*  simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]" *)
+    "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
+
+defs
+  is_gcd_def  "is_gcd p m n == p dvd m  &  p dvd n  &
+                               (ALL d. d dvd m & d dvd n --> d dvd p)"
+
+  coprime_def "coprime m n == gcd(m,n) = 1"
+
+  prime_def   "prime == {p. 1<p & (ALL m. m dvd p --> m=1 | m=p)}"
+
+end
diff --git a/etc/isa/depends/Usedepends.ML b/etc/isa/depends/Usedepends.ML
new file mode 100644
index 00000000..7557d6e8
--- /dev/null
+++ b/etc/isa/depends/Usedepends.ML
@@ -0,0 +1,3 @@
+use "~/ProofGeneral/isa/depends.ML";
+
+
diff --git a/etc/isa/depends/Usedepends.thy b/etc/isa/depends/Usedepends.thy
new file mode 100644
index 00000000..4f8eb516
--- /dev/null
+++ b/etc/isa/depends/Usedepends.thy
@@ -0,0 +1,5 @@
+(* dummy theory to load depends.ML *)
+theory Usedepends = Main:
+end
+  
+