src/HOL/ex/Primrec.ML

 *)
 
 
 (** Useful special cases of evaluation ***)
 
-goalw thy [SC_def] "SC (x#l) = Suc x";
+Goalw [SC_def] "SC (x#l) = Suc x";
 by (Asm_simp_tac 1);
 qed "SC";
 
-goalw thy [CONST_def] "CONST k l = k";
+Goalw [CONST_def] "CONST k l = k";
 by (Asm_simp_tac 1);
 qed "CONST";
 
-goalw thy [PROJ_def] "PROJ(0) (x#l) = x";
+Goalw [PROJ_def] "PROJ(0) (x#l) = x";
 by (Asm_simp_tac 1);
 qed "PROJ_0";
 
-goalw thy [COMP_def] "COMP g [f] l = g [f l]";
+Goalw [COMP_def] "COMP g [f] l = g [f l]";
 by (Asm_simp_tac 1);
 qed "COMP_1";
 
-goalw thy [PREC_def] "PREC f g (0#l) = f l";
+Goalw [PREC_def] "PREC f g (0#l) = f l";
 by (Asm_simp_tac 1);
 qed "PREC_0";
 
-goalw thy [PREC_def] "PREC f g (Suc x # l) = g (PREC f g (x#l) # x # l)";
+Goalw [PREC_def] "PREC f g (Suc x # l) = g (PREC f g (x#l) # x # l)";
 by (Asm_simp_tac 1);
 qed "PREC_Suc";
 
 Addsimps [SC, CONST, PROJ_0, COMP_1, PREC_0, PREC_Suc];
 
 
 Addsimps ack.rules;
 
 (*PROPERTY A 4*)
-goal thy "j < ack(i,j)";
+Goal "j < ack(i,j)";
 by (res_inst_tac [("u","i"),("v","j")] ack.induct 1);
 by (ALLGOALS Asm_simp_tac);
 by (ALLGOALS trans_tac);
 qed "less_ack2";
 
 AddIffs [less_ack2];
 
 (*PROPERTY A 5-, the single-step lemma*)
-goal thy "ack(i,j) < ack(i, Suc(j))";
+Goal "ack(i,j) < ack(i, Suc(j))";
 by (res_inst_tac [("u","i"),("v","j")] ack.induct 1);
 by (ALLGOALS Asm_simp_tac);
 qed "ack_less_ack_Suc2";
 
 AddIffs [ack_less_ack_Suc2];
 
 (*PROPERTY A 5, monotonicity for < *)
-goal thy "j<k --> ack(i,j) < ack(i,k)";
+Goal "j<k --> ack(i,j) < ack(i,k)";
 by (res_inst_tac [("u","i"),("v","k")] ack.induct 1);
 by (ALLGOALS Asm_simp_tac);
 by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
 qed_spec_mp "ack_less_mono2";
 
 (*PROPERTY A 5', monotonicity for<=*)
-goal thy "!!i j k. j<=k ==> ack(i,j)<=ack(i,k)";
+Goal "!!i j k. j<=k ==> ack(i,j)<=ack(i,k)";
 by (full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
 by (blast_tac (claset() addIs [ack_less_mono2]) 1);
 qed "ack_le_mono2";
 
 (*PROPERTY A 6*)
-goal thy "ack(i, Suc(j)) <= ack(Suc(i), j)";
+Goal "ack(i, Suc(j)) <= ack(Suc(i), j)";
 by (induct_tac "j" 1);
 by (ALLGOALS Asm_simp_tac);
 by (blast_tac (claset() addIs [ack_le_mono2, less_ack2 RS Suc_leI, 
 			      le_trans]) 1);
 qed "ack2_le_ack1";
 
 AddIffs [ack2_le_ack1];
 
 (*PROPERTY A 7-, the single-step lemma*)
-goal thy "ack(i,j) < ack(Suc(i),j)";
+Goal "ack(i,j) < ack(Suc(i),j)";
 by (blast_tac (claset() addIs [ack_less_mono2, less_le_trans]) 1);
 qed "ack_less_ack_Suc1";
 
 AddIffs [ack_less_ack_Suc1];
 
 (*PROPERTY A 4'? Extra lemma needed for CONST case, constant functions*)
-goal thy "i < ack(i,j)";
+Goal "i < ack(i,j)";
 by (induct_tac "i" 1);
 by (ALLGOALS Asm_simp_tac);
 by (blast_tac (claset() addIs [Suc_leI, le_less_trans]) 1);
 qed "less_ack1";
 AddIffs [less_ack1];
 
 (*PROPERTY A 8*)
-goal thy "ack(1,j) = Suc(Suc(j))";
+Goal "ack(1,j) = Suc(Suc(j))";
 by (induct_tac "j" 1);
 by (ALLGOALS Asm_simp_tac);
 qed "ack_1";
 Addsimps [ack_1];
 
 (*PROPERTY A 9*)
-goal thy "ack(Suc(1),j) = Suc(Suc(Suc(j+j)))";
+Goal "ack(Suc(1),j) = Suc(Suc(Suc(j+j)))";
 by (induct_tac "j" 1);
 by (ALLGOALS Asm_simp_tac);
 qed "ack_2";
 Addsimps [ack_2];
 
 
 (*PROPERTY A 7, monotonicity for < [not clear why ack_1 is now needed first!]*)
-goal thy "ack(i,k) < ack(Suc(i+i'),k)";
+Goal "ack(i,k) < ack(Suc(i+i'),k)";
 by (res_inst_tac [("u","i"),("v","k")] ack.induct 1);
 by (ALLGOALS Asm_full_simp_tac);
 by (blast_tac (claset() addIs [less_trans, ack_less_mono2]) 2);
 by (res_inst_tac [("u","i'"),("v","n")] ack.induct 1);
 by (ALLGOALS Asm_simp_tac);
 by (blast_tac (claset() addIs [less_trans, ack_less_mono2]) 1);
 by (blast_tac (claset() addIs [Suc_leI RS le_less_trans, ack_less_mono2]) 1);
 val lemma = result();
 
-goal thy "!!i j k. i<j ==> ack(i,k) < ack(j,k)";
+Goal "!!i j k. i<j ==> ack(i,k) < ack(j,k)";
 by (etac less_natE 1);
 by (blast_tac (claset() addSIs [lemma]) 1);
 qed "ack_less_mono1";
 
 (*PROPERTY A 7', monotonicity for<=*)
-goal thy "!!i j k. i<=j ==> ack(i,k)<=ack(j,k)";
+Goal "!!i j k. i<=j ==> ack(i,k)<=ack(j,k)";
 by (full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
 by (blast_tac (claset() addIs [ack_less_mono1]) 1);
 qed "ack_le_mono1";
 
 (*PROPERTY A 10*)
-goal thy "ack(i1, ack(i2,j)) < ack(Suc(Suc(i1+i2)), j)";
+Goal "ack(i1, ack(i2,j)) < ack(Suc(Suc(i1+i2)), j)";
 by (rtac (ack2_le_ack1 RSN (2,less_le_trans)) 1);
 by (Asm_simp_tac 1);
 by (rtac (le_add1 RS ack_le_mono1 RS le_less_trans) 1);
 by (rtac (ack_less_mono1 RS ack_less_mono2) 1);
 by (simp_tac (simpset() addsimps [le_imp_less_Suc, le_add2]) 1);
 qed "ack_nest_bound";
 
 (*PROPERTY A 11*)
-goal thy "ack(i1,j) + ack(i2,j) < ack(Suc(Suc(Suc(Suc(i1+i2)))), j)";
+Goal "ack(i1,j) + ack(i2,j) < ack(Suc(Suc(Suc(Suc(i1+i2)))), j)";
 by (res_inst_tac [("j", "ack(Suc(1), ack(i1 + i2, j))")] less_trans 1);
 by (Asm_simp_tac 1);
 by (rtac (ack_nest_bound RS less_le_trans) 2);
 by (Asm_simp_tac 2);
 by (blast_tac (claset() addSIs [le_add1, le_add2]

 			       add_le_mono]) 1);
 qed "ack_add_bound";
 
 (*PROPERTY A 12.  Article uses existential quantifier but the ALF proof
   used k+4.  Quantified version must be nested EX k'. ALL i,j... *)
-goal thy "!!i j k. i < ack(k,j) ==> i+j < ack(Suc(Suc(Suc(Suc(k)))), j)";
+Goal "!!i j k. i < ack(k,j) ==> i+j < ack(Suc(Suc(Suc(Suc(k)))), j)";
 by (res_inst_tac [("j", "ack(k,j) + ack(0,j)")] less_trans 1);
 by (rtac (ack_add_bound RS less_le_trans) 2);
 by (Asm_simp_tac 2);
 by (REPEAT (ares_tac ([add_less_mono, less_ack2]) 1));
 qed "ack_add_bound2";

 
 (*** Inductive definition of the PR functions ***)
 
 (*** MAIN RESULT ***)
 
-goalw thy [SC_def] "SC l < ack(1, list_add l)";
+Goalw [SC_def] "SC l < ack(1, list_add l)";
 by (induct_tac "l" 1);
 by (ALLGOALS (simp_tac (simpset() addsimps [le_add1, le_imp_less_Suc])));
 qed "SC_case";
 
-goal thy "CONST k l < ack(k, list_add l)";
+Goal "CONST k l < ack(k, list_add l)";
 by (Simp_tac 1);
 qed "CONST_case";
 
-goalw thy [PROJ_def] "ALL i. PROJ i l < ack(0, list_add l)";
+Goalw [PROJ_def] "ALL i. PROJ i l < ack(0, list_add l)";
 by (Simp_tac 1);
 by (induct_tac "l" 1);
 by (ALLGOALS Asm_simp_tac);
 by (rtac allI 1);
 by (exhaust_tac "i" 1);

                        addSIs [le_add2]) 1);
 qed_spec_mp "PROJ_case";
 
 (** COMP case **)
 
-goal thy
+Goal
  "!!fs. fs : lists(PRIMREC Int {f. EX kf. ALL l. f l < ack(kf, list_add l)}) \
 \      ==> EX k. ALL l. list_add (map(%f. f l) fs) < ack(k, list_add l)";
 by (etac lists.induct 1);
 by (res_inst_tac [("x","0")] exI 1 THEN Asm_simp_tac 1);
 by Safe_tac;
 by (Asm_simp_tac 1);
 by (blast_tac (claset() addIs [add_less_mono, ack_add_bound, less_trans]) 1);
 qed "COMP_map_lemma";
 
-goalw thy [COMP_def]
+Goalw [COMP_def]
  "!!g. [| ALL l. g l < ack(kg, list_add l);           \
 \         fs: lists(PRIMREC Int {f. EX kf. ALL l. f l < ack(kf, list_add l)}) \
 \      |] ==> EX k. ALL l. COMP g fs  l < ack(k, list_add l)";
 by (forward_tac [impOfSubs (Int_lower1 RS lists_mono)] 1);
 by (etac (COMP_map_lemma RS exE) 1);

 by (blast_tac (claset() addIs [ack_less_mono2, ack_nest_bound, less_trans]) 1);
 qed "COMP_case";
 
 (** PREC case **)
 
-goalw thy [PREC_def]
+Goalw [PREC_def]
  "!!f g. [| ALL l. f l + list_add l < ack(kf, list_add l); \
 \           ALL l. g l + list_add l < ack(kg, list_add l)  \
 \        |] ==> PREC f g l + list_add l < ack(Suc(kf+kg), list_add l)";
 by (exhaust_tac "l" 1);
 by (ALLGOALS Asm_simp_tac);

 by (Asm_simp_tac 1);
 by (rtac (le_add2 RS ack_le_mono1 RS le_less_trans) 1);
 by (etac ack_less_mono2 1);
 qed "PREC_case_lemma";
 
-goal thy
+Goal
  "!!f g. [| ALL l. f l < ack(kf, list_add l);        \
 \           ALL l. g l < ack(kg, list_add l)         \
 \        |] ==> EX k. ALL l. PREC f g l< ack(k, list_add l)";
 by (rtac exI 1);
 by (rtac allI 1);
 by (rtac ([le_add1, PREC_case_lemma] MRS le_less_trans) 1);
 by (REPEAT (blast_tac (claset() addIs [ack_add_bound2]) 1));
 qed "PREC_case";
 
-goal thy "!!f. f:PRIMREC ==> EX k. ALL l. f l < ack(k, list_add l)";
+Goal "!!f. f:PRIMREC ==> EX k. ALL l. f l < ack(k, list_add l)";
 by (etac PRIMREC.induct 1);
 by (ALLGOALS 
     (blast_tac (claset() addIs [SC_case, CONST_case, PROJ_case, COMP_case, 
 			       PREC_case])));
 qed "ack_bounds_PRIMREC";
 
-goal thy "(%l. case l of [] => 0 | x#l' => ack(x,x)) ~: PRIMREC";
+Goal "(%l. case l of [] => 0 | x#l' => ack(x,x)) ~: PRIMREC";
 by (rtac notI 1);
 by (etac (ack_bounds_PRIMREC RS exE) 1);
 by (rtac less_irrefl 1);
 by (dres_inst_tac [("x", "[x]")] spec 1);
 by (Asm_full_simp_tac 1);