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(* Title: HOL/MiniML/W.ML
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ID: $Id$
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Author: Dieter Nazareth and Tobias Nipkow
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Copyright 1995 TU Muenchen
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Correctness and completeness of type inference algorithm W
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*)
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open W;
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(* stronger version of Suc_leD *)
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goalw Nat.thy [le_def]
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"!!m. Suc m <= n ==> m < n";
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1300
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by (Asm_full_simp_tac 1);
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by (cut_facts_tac [less_linear] 1);
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by (fast_tac HOL_cs 1);
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qed "Suc_le_lessD";
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Addsimps [Suc_le_lessD];
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(* correctness of W with respect to has_type *)
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goal thy
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"!a s t m n . Ok (s,t,m) = W e a n --> ($ s a |- e :: t)";
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1300
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by (expr.induct_tac "e" 1);
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(* case Var n *)
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by (asm_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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(* case Abs e *)
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by (asm_full_simp_tac (!simpset addsimps [app_subst_list]
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setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (eres_inst_tac [("x","TVar(n) # a")] allE 1);
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by( fast_tac (HOL_cs addss (!simpset addsimps [eq_sym_conv])) 1);
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(* case App e1 e2 *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by( rename_tac "sa ta na sb tb nb sc" 1);
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by (res_inst_tac [("t2.0","$ sc tb")] has_type.AppI 1);
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by (res_inst_tac [("s1","sc")] (app_subst_TVar RS subst) 1);
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by (rtac (app_subst_Fun RS subst) 1);
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by (res_inst_tac [("t","$ sc (tb -> (TVar nb))"),("s","$ sc ($ sb ta)")] subst 1);
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by (Asm_full_simp_tac 1);
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by (simp_tac (HOL_ss addsimps [subst_comp_tel RS sym]) 1);
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by ( (rtac has_type_cl_sub 1) THEN (rtac has_type_cl_sub 1));
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by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
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by (asm_full_simp_tac (!simpset addsimps [subst_comp_tel RS sym,has_type_cl_sub,eq_sym_conv]) 1);
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qed "W_correct";
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val has_type_casesE = map(has_type.mk_cases expr.simps)
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[" s |- Var n :: t"," s |- Abs e :: t","s |- App e1 e2 ::t"];
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(* the resulting type variable is always greater or equal than the given one *)
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goal thy
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"!a n s t m. W e a n = Ok (s,t,m) --> n<=m";
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by (expr.induct_tac "e" 1);
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(* case Var(n) *)
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by (fast_tac (HOL_cs addss (!simpset setloop (split_tac [expand_if]))) 1);
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(* case Abs e *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (fast_tac (HOL_cs addDs [Suc_leD]) 1);
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(* case App e1 e2 *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (rename_tac "s t na sa ta nb sb sc tb m" 1);
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by (eres_inst_tac [("x","a")] allE 1);
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by (eres_inst_tac [("x","n")] allE 1);
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by (eres_inst_tac [("x","$ s a")] allE 1);
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by (eres_inst_tac [("x","s")] allE 1);
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by (eres_inst_tac [("x","t")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
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by (etac conjE 1);
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by (eres_inst_tac [("x","sa")] allE 1);
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by (eres_inst_tac [("x","ta")] allE 1);
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by (eres_inst_tac [("x","nb")] allE 1);
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by (etac conjE 1);
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by (res_inst_tac [("j","na")] le_trans 1);
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by (Asm_simp_tac 1);
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by (asm_simp_tac (!simpset addsimps [Suc_leD]) 1);
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qed "W_var_ge";
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Addsimps [W_var_ge];
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goal thy
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"!! s. Ok (s,t,m) = W e a n ==> n<=m";
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by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
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qed "W_var_geD";
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(* auxiliary lemma *)
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goal Maybe.thy "(y = Ok x) = (Ok x = y)";
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by( simp_tac (!simpset addsimps [eq_sym_conv]) 1);
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qed "rotate_Ok";
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(* resulting type variable is new *)
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goal thy
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"!n a s t m. new_tv n a --> W e a n = Ok (s,t,m) --> \
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\ (new_tv m s & new_tv m t)";
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by (expr.induct_tac "e" 1);
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(* case Var n *)
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by (fast_tac (HOL_cs addss (!simpset
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addsimps [id_subst_def,list_all_nth,new_tv_list,new_tv_subst]
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setloop (split_tac [expand_if]))) 1);
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(* case Abs e *)
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by (simp_tac (!simpset addsimps [new_tv_subst,new_tv_Suc_list]
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setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (eres_inst_tac [("x","Suc n")] allE 1);
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by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
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by (fast_tac (HOL_cs addss (!simpset
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addsimps [new_tv_subst,new_tv_Suc_list])) 1);
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(* case App e1 e2 *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (rename_tac "s t na sa ta nb sb sc tb m" 1);
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by (eres_inst_tac [("x","n")] allE 1);
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by (eres_inst_tac [("x","a")] allE 1);
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by (eres_inst_tac [("x","s")] allE 1);
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by (eres_inst_tac [("x","t")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
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by (eres_inst_tac [("x","$ s a")] allE 1);
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by (eres_inst_tac [("x","sa")] allE 1);
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by (eres_inst_tac [("x","ta")] allE 1);
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by (eres_inst_tac [("x","nb")] allE 1);
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by( asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
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by (rtac conjI 1);
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by (rtac new_tv_subst_comp_2 1);
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by (rtac new_tv_subst_comp_2 1);
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by (rtac (lessI RS less_imp_le RS new_tv_subst_le) 1);
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by (res_inst_tac [("n","na")] new_tv_subst_le 1);
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by (asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
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by (Asm_simp_tac 1);
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by (fast_tac (HOL_cs addDs [W_var_geD] addIs
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[new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS new_tv_subst_le])
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1);
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by (etac (sym RS mgu_new) 1);
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by (fast_tac (HOL_cs addDs [W_var_geD] addIs [new_tv_subst_te,new_tv_list_le,
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new_tv_subst_tel,lessI RS less_imp_le RS new_tv_le,lessI RS less_imp_le RS
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new_tv_subst_le,new_tv_le]) 1);
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by (fast_tac (HOL_cs addDs [W_var_geD] addIs
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[new_tv_list_le,new_tv_subst_tel,new_tv_le]
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addss (!simpset)) 1);
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by (rtac (lessI RS new_tv_subst_var) 1);
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by (etac (sym RS mgu_new) 1);
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by (fast_tac (HOL_cs addSIs [lessI RS less_imp_le RS new_tv_le,new_tv_subst_te]
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addDs [W_var_geD] addIs
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[new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS
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new_tv_subst_le,new_tv_le] addss !simpset) 1);
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by (fast_tac (HOL_cs addDs [W_var_geD] addIs
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[new_tv_list_le,new_tv_subst_tel,new_tv_le]
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addss (!simpset)) 1);
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bind_thm ("new_tv_W",result() RS spec RS spec RS spec RS spec RS spec RS mp RS mp);
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goal thy
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"!n a s t m v. W e a n = Ok (s,t,m) --> \
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\ (v:free_tv s | v:free_tv t) --> v<n --> v:free_tv a";
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by (expr.induct_tac "e" 1);
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(* case Var n *)
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by (fast_tac (HOL_cs addIs [nth_mem,subsetD,ftv_mem_sub_ftv_list]
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addss (!simpset setloop (split_tac [expand_if]))) 1);
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(* case Abs e *)
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by (asm_full_simp_tac (!simpset addsimps
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[free_tv_subst] setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (rename_tac "s t na sa ta m v" 1);
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by (eres_inst_tac [("x","Suc n")] allE 1);
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by (eres_inst_tac [("x","TVar n # a")] allE 1);
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by (eres_inst_tac [("x","s")] allE 1);
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by (eres_inst_tac [("x","t")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (eres_inst_tac [("x","v")] allE 1);
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by (fast_tac (HOL_cs addIs [cod_app_subst] addss !simpset) 1);
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(* case App e1 e2 *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (rename_tac "s t na sa ta nb sb sc tb m v" 1);
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by (eres_inst_tac [("x","n")] allE 1);
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by (eres_inst_tac [("x","a")] allE 1);
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by (eres_inst_tac [("x","s")] allE 1);
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by (eres_inst_tac [("x","t")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (eres_inst_tac [("x","na")] allE 1);
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by (eres_inst_tac [("x","v")] allE 1);
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(* second case *)
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by (eres_inst_tac [("x","$ s a")] allE 1);
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by (eres_inst_tac [("x","sa")] allE 1);
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by (eres_inst_tac [("x","ta")] allE 1);
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by (eres_inst_tac [("x","nb")] allE 1);
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by (eres_inst_tac [("x","v")] allE 1);
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by (safe_tac (empty_cs addSIs [conjI,impI] addSEs [conjE]) );
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by (asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
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by (dtac W_var_geD 1);
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by (dtac W_var_geD 1);
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by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
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by (fast_tac (HOL_cs addDs [free_tv_comp_subst RS subsetD,sym RS mgu_free,
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codD,free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD,
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less_le_trans,less_not_refl2,subsetD]
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addEs [UnE]
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addss !simpset) 1);
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by (Asm_full_simp_tac 1);
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by (dtac (sym RS W_var_geD) 1);
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by (dtac (sym RS W_var_geD) 1);
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by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
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by (fast_tac (HOL_cs addDs [mgu_free, codD,free_tv_subst_var RS subsetD,
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free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD,
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less_le_trans,subsetD]
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addEs [UnE]
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addss !simpset) 1);
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bind_thm ("free_tv_W",result() RS spec RS spec RS spec RS spec RS spec RS spec RS mp RS mp RS mp);
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goal HOL.thy "(~(P | Q)) = (~P & ~Q)";
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by( fast_tac HOL_cs 1);
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qed "not_disj";
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(* Completeness of W w.r.t. has_type *)
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goal thy
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"!s' a t' n. ($ s' a |- e :: t') --> new_tv n a --> \
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\ (? s t. (? m. W e a n = Ok (s,t,m) ) & \
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\ (? r. $ s' a = $ r ($ s a) & \
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\ t' = $ r t ) )";
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by (expr.induct_tac "e" 1);
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(* case Var n *)
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by (strip_tac 1);
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by (simp_tac (!simpset addcongs [conj_cong]
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setloop (split_tac [expand_if])) 1);
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by (eresolve_tac has_type_casesE 1);
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by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv,app_subst_list]) 1);
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by (res_inst_tac [("x","id_subst")] exI 1);
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by (res_inst_tac [("x","nth nat a")] exI 1);
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by (Simp_tac 1);
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by (res_inst_tac [("x","s'")] exI 1);
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by (Asm_simp_tac 1);
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(* case Abs e *)
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by (strip_tac 1);
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by (eresolve_tac has_type_casesE 1);
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by (eres_inst_tac [("x","%x.if x=n then t1 else (s' x)")] allE 1);
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by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
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by (eres_inst_tac [("x","t2")] allE 1);
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by (eres_inst_tac [("x","Suc n")] allE 1);
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by (fast_tac (HOL_cs addss (!simpset addcongs [conj_cong]
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setloop (split_tac [expand_bind]))) 1);
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(* case App e1 e2 *)
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by (strip_tac 1);
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by (eresolve_tac has_type_casesE 1);
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by (eres_inst_tac [("x","s'")] allE 1);
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by (eres_inst_tac [("x","a")] allE 1);
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1400
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by (eres_inst_tac [("x","t2 -> t'")] allE 1);
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1300
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by (eres_inst_tac [("x","n")] allE 1);
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by (safe_tac HOL_cs);
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by (eres_inst_tac [("x","r")] allE 1);
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by (eres_inst_tac [("x","$ s a")] allE 1);
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by (eres_inst_tac [("x","t2")] allE 1);
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by (eres_inst_tac [("x","m")] allE 1);
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1465
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by (dtac asm_rl 1);
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by (dtac asm_rl 1);
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by (dtac asm_rl 1);
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1300
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by (Asm_full_simp_tac 1);
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by (safe_tac HOL_cs);
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by (fast_tac HOL_cs 1);
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by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS
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conjunct1,new_tv_list_le,new_tv_subst_tel]) 1);
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1300
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by (subgoal_tac
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"$ (%x.if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
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\ else ra x)) ($ sa t) = \
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\ $ (%x.if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
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1400
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\ else ra x)) (ta -> (TVar ma))" 1);
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1300
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by (res_inst_tac [("t","$ (%x. if x = ma then t' else \
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\ (if x:(free_tv t - free_tv sa) then r x else ra x)) ($ sa t)"),
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1400
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("s","($ ra ta) -> t'")] ssubst 2);
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1300
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by (asm_simp_tac (!simpset addsimps [subst_comp_te]) 2);
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1465
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by (rtac eq_free_eq_subst_te 2);
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1300
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by (strip_tac 2);
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by (subgoal_tac "na ~=ma" 2);
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by (fast_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD,
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287 |
new_tv_not_free_tv,new_tv_le]) 3);
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|
288 |
by (case_tac "na:free_tv sa" 2);
|
|
289 |
(* na ~: free_tv sa *)
|
|
290 |
by (asm_simp_tac (!simpset addsimps [not_free_impl_id]
|
|
291 |
setloop (split_tac [expand_if])) 3);
|
|
292 |
(* na : free_tv sa *)
|
1400
|
293 |
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
|
1465
|
294 |
by (dtac eq_subst_tel_eq_free 2);
|
1300
|
295 |
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
|
|
296 |
by (Asm_simp_tac 2);
|
|
297 |
by (case_tac "na:dom sa" 2);
|
|
298 |
(* na ~: dom sa *)
|
|
299 |
by (asm_full_simp_tac (!simpset addsimps [dom_def]
|
|
300 |
setloop (split_tac [expand_if])) 3);
|
|
301 |
(* na : dom sa *)
|
1465
|
302 |
by (rtac eq_free_eq_subst_te 2);
|
1300
|
303 |
by (strip_tac 2);
|
|
304 |
by (subgoal_tac "nb ~= ma" 2);
|
|
305 |
by ((forward_tac [new_tv_W] 3) THEN (atac 3));
|
1465
|
306 |
by (etac conjE 3);
|
|
307 |
by (dtac new_tv_subst_tel 3);
|
1300
|
308 |
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 3);
|
|
309 |
by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss
|
|
310 |
(!simpset addsimps [cod_def,free_tv_subst])) 3);
|
|
311 |
by (fast_tac (set_cs addss (!simpset addsimps [cod_def,free_tv_subst]
|
|
312 |
setloop (split_tac [expand_if]))) 2);
|
|
313 |
|
|
314 |
by (Simp_tac 2);
|
1465
|
315 |
by (rtac eq_free_eq_subst_te 2);
|
1300
|
316 |
by (strip_tac 2 );
|
|
317 |
by (subgoal_tac "na ~= ma" 2);
|
|
318 |
by ((forward_tac [new_tv_W] 3) THEN (atac 3));
|
1465
|
319 |
by (etac conjE 3);
|
|
320 |
by (dtac (sym RS W_var_geD) 3);
|
1300
|
321 |
by (fast_tac (HOL_cs addDs [new_tv_list_le,new_tv_subst_tel,new_tv_W,new_tv_not_free_tv]) 3);
|
|
322 |
by (case_tac "na: free_tv t - free_tv sa" 2);
|
|
323 |
(* case na ~: free_tv t - free_tv sa *)
|
|
324 |
by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 3);
|
|
325 |
(* case na : free_tv t - free_tv sa *)
|
|
326 |
by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
|
1400
|
327 |
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
|
1465
|
328 |
by (dtac eq_subst_tel_eq_free 2);
|
1300
|
329 |
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
|
|
330 |
by (asm_full_simp_tac (!simpset addsimps [free_tv_subst,dom_def,not_disj]) 2);
|
|
331 |
|
|
332 |
by (asm_simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
|
|
333 |
by (safe_tac HOL_cs );
|
1465
|
334 |
by (dtac mgu_Ok 1);
|
1300
|
335 |
by( fast_tac (HOL_cs addss !simpset) 1);
|
|
336 |
by (REPEAT (resolve_tac [exI,conjI] 1));
|
|
337 |
by (fast_tac HOL_cs 1);
|
|
338 |
by (fast_tac HOL_cs 1);
|
|
339 |
by ((dtac mgu_mg 1) THEN (atac 1));
|
1465
|
340 |
by (etac exE 1);
|
1300
|
341 |
by (res_inst_tac [("x","rb")] exI 1);
|
1465
|
342 |
by (rtac conjI 1);
|
1300
|
343 |
by (dres_inst_tac [("x","ma")] fun_cong 2);
|
|
344 |
by( asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 2);
|
|
345 |
by (simp_tac (!simpset addsimps [subst_comp_tel RS sym]) 1);
|
1400
|
346 |
by (res_inst_tac [("ts2","($ sa ($ s a))")] ((subst_comp_tel RS sym) RSN
|
1300
|
347 |
(2,trans)) 1);
|
|
348 |
by( asm_full_simp_tac (!simpset addsimps [o_def,eq_sym_conv]) 1);
|
1465
|
349 |
by (rtac eq_free_eq_subst_tel 1);
|
1300
|
350 |
by( safe_tac HOL_cs );
|
|
351 |
by (subgoal_tac "ma ~= na" 1);
|
|
352 |
by ((forward_tac [new_tv_W] 2) THEN (atac 2));
|
1465
|
353 |
by (etac conjE 2);
|
|
354 |
by (dtac new_tv_subst_tel 2);
|
1300
|
355 |
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 2);
|
|
356 |
by (( forw_inst_tac [("xd","m")] (sym RSN (2,new_tv_W)) 2) THEN (atac 2));
|
1465
|
357 |
by (etac conjE 2);
|
|
358 |
by (dtac (free_tv_app_subst_tel RS subsetD) 2);
|
1300
|
359 |
by (fast_tac (set_cs addDs [W_var_geD,new_tv_list_le,codD,
|
|
360 |
new_tv_not_free_tv]) 2);
|
|
361 |
by (case_tac "na: free_tv t - free_tv sa" 1);
|
|
362 |
(* case na ~: free_tv t - free_tv sa *)
|
|
363 |
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
|
|
364 |
(* case na : free_tv t - free_tv sa *)
|
|
365 |
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
|
1465
|
366 |
by (dtac (free_tv_app_subst_tel RS subsetD) 1);
|
1300
|
367 |
by (fast_tac (set_cs addDs [codD,subst_comp_tel RSN (2,trans),
|
|
368 |
eq_subst_tel_eq_free] addss ((!simpset addsimps
|
|
369 |
[not_disj,free_tv_subst,dom_def]))) 1);
|
|
370 |
qed "W_complete";
|
|
371 |
|
|
372 |
|
|
373 |
|
|
374 |
|
|
375 |
|
|
376 |
|
|
377 |
|
|
378 |
|
|
379 |
|