src/HOL/MiniML/W.ML
author nipkow
Wed, 22 May 1996 17:11:54 +0200
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permissions -rw-r--r--
Added the second half of the W/I correspondence.
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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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by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 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 W.thy
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        "!a s t m n . Ok (s,t,m) = W e a n --> $s a |- e :: t";
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by (expr.induct_tac "e" 1);
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(* case Var n *)
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by (asm_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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(* case Abs e *)
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by (asm_full_simp_tac (!simpset addsimps [app_subst_list]
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                        setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by (eres_inst_tac [("x","TVar(n) # a")] allE 1);
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by( fast_tac (HOL_cs addss (!simpset addsimps [eq_sym_conv])) 1);
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(* case App e1 e2 *)
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by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
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by (strip_tac 1);
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by( rename_tac "sa ta na sb tb nb sc" 1);
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by (res_inst_tac [("t2.0","$ 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,o_def,has_type_cl_sub,eq_sym_conv]) 1);
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qed_spec_mp "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 1);
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qed_spec_mp "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 [o_def,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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qed_spec_mp "new_tv_W";
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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]
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                     addss (!simpset addsimps [less_Suc_eq])) 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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   196
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,o_def]) 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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   211
by (dtac (sym RS W_var_geD) 1);
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   212
by (dtac (sym RS W_var_geD) 1);
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   213
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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7b95d7b49f7a Introduced qed_spec_mp.
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qed_spec_mp "free_tv_W"; 
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c7a8f374339b New theory: type inference for let-free MiniML
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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)) &  \
d127436567d0 modified priorities in syntax
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\                     (? r. $s' a = $r ($s a) & t' = $r t))";
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by (expr.induct_tac "e" 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   227
(* case Var n *)
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   228
by (strip_tac 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   229
by (simp_tac (!simpset addcongs [conj_cong] 
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    setloop (split_tac [expand_if])) 1);
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   231
by (eresolve_tac has_type_casesE 1); 
c7a8f374339b New theory: type inference for let-free MiniML
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   232
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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   234
by (res_inst_tac [("x","nth nat a")] exI 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   235
by (Simp_tac 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   236
by (res_inst_tac [("x","s'")] exI 1);
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   237
by (Asm_simp_tac 1);
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(* case Abs e *)
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by (strip_tac 1);
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   241
by (eresolve_tac has_type_casesE 1);
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   242
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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   246
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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   251
by (eresolve_tac has_type_casesE 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   252
by (eres_inst_tac [("x","s'")] allE 1);
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   253
by (eres_inst_tac [("x","a")] allE 1);
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5d909faf0e04 Introduced Monad syntax Pat := Val; Cont
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   254
by (eres_inst_tac [("x","t2 -> t'")] allE 1);
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by (eres_inst_tac [("x","n")] allE 1);
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   256
by (safe_tac HOL_cs);
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by (eres_inst_tac [("x","r")] allE 1);
c7a8f374339b New theory: type inference for let-free MiniML
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by (eres_inst_tac [("x","$ s a")] allE 1);
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   259
by (eres_inst_tac [("x","t2")] allE 1);
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   260
by (eres_inst_tac [("x","m")] allE 1);
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   261
by (dtac asm_rl 1);
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   262
by (dtac asm_rl 1);
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by (dtac asm_rl 1);
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c7a8f374339b New theory: type inference for let-free MiniML
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by (Asm_full_simp_tac 1);
c7a8f374339b New theory: type inference for let-free MiniML
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   265
by (safe_tac HOL_cs);
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   266
by (fast_tac HOL_cs 1);
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   267
by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS
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   268
        conjunct1,new_tv_list_le,new_tv_subst_tel]) 1);
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c7a8f374339b New theory: type inference for let-free MiniML
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   270
by (subgoal_tac
c7a8f374339b New theory: type inference for let-free MiniML
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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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5d909faf0e04 Introduced Monad syntax Pat := Val; Cont
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   274
\        else ra x)) (ta -> (TVar ma))" 1);
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by (res_inst_tac [("t","$ (%x. if x = ma then t' else \
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   276
\   (if x:(free_tv t - free_tv sa) then r x else ra x)) ($ sa t)"),
1400
5d909faf0e04 Introduced Monad syntax Pat := Val; Cont
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   277
    ("s","($ ra ta) -> t'")] ssubst 2);
1300
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parents:
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   278
by (asm_simp_tac (!simpset addsimps [subst_comp_te]) 2);
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   279
by (rtac eq_free_eq_subst_te 2);  
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   280
by (strip_tac 2);
c7a8f374339b New theory: type inference for let-free MiniML
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parents:
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   281
by (subgoal_tac "na ~=ma" 2);
c7a8f374339b New theory: type inference for let-free MiniML
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parents:
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   282
by (fast_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD,
c7a8f374339b New theory: type inference for let-free MiniML
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parents:
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   283
    new_tv_not_free_tv,new_tv_le]) 3);
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parents:
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   284
by (case_tac "na:free_tv sa" 2);
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   285
(* na ~: free_tv sa *)
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by (asm_simp_tac (!simpset addsimps [not_free_impl_id]
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    setloop (split_tac [expand_if])) 3);
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(* na : free_tv sa *)
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by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
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by (dtac eq_subst_tel_eq_free 2);
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by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
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by (Asm_simp_tac 2); 
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by (case_tac "na:dom sa" 2);
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(* na ~: dom sa *)
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by (asm_full_simp_tac (!simpset addsimps [dom_def] 
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    setloop (split_tac [expand_if])) 3);
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(* na : dom sa *)
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by (rtac eq_free_eq_subst_te 2);
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by (strip_tac 2);
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by (subgoal_tac "nb ~= ma" 2);
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by ((forward_tac [new_tv_W] 3) THEN (atac 3));
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by (etac conjE 3);
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by (dtac new_tv_subst_tel 3);
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by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 3);
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by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss 
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    (!simpset addsimps [cod_def,free_tv_subst])) 3);
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by (fast_tac (set_cs addss (!simpset addsimps [cod_def,free_tv_subst] 
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    setloop (split_tac [expand_if]))) 2);
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by (Simp_tac 2);  
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by (rtac eq_free_eq_subst_te 2);
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by (strip_tac 2 );
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by (subgoal_tac "na ~= ma" 2);
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by ((forward_tac [new_tv_W] 3) THEN (atac 3));
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by (etac conjE 3);
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by (dtac (sym RS W_var_geD) 3);
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by (fast_tac (HOL_cs addDs [new_tv_list_le,new_tv_subst_tel,new_tv_W,new_tv_not_free_tv]) 3);
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by (case_tac "na: free_tv t - free_tv sa" 2);
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(* case na ~: free_tv t - free_tv sa *)
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by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 3);
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(* case na : free_tv t - free_tv sa *)
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by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
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by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
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by (dtac eq_subst_tel_eq_free 2);
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by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
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by (asm_full_simp_tac (!simpset addsimps [free_tv_subst,dom_def,de_Morgan_disj]) 2);
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by (asm_simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 
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by (safe_tac HOL_cs );
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by (dtac mgu_Ok 1);
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by( fast_tac (HOL_cs addss !simpset) 1);
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by (REPEAT (resolve_tac [exI,conjI] 1));
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by (fast_tac HOL_cs 1);
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by (fast_tac HOL_cs 1);
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by ((dtac mgu_mg 1) THEN (atac 1));
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by (etac exE 1);
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by (res_inst_tac [("x","rb")] exI 1);
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by (rtac conjI 1);
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by (dres_inst_tac [("x","ma")] fun_cong 2);
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by( asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 2);
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by (simp_tac (!simpset addsimps [subst_comp_tel RS sym]) 1);
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by (res_inst_tac [("ts2","($ sa ($ s a))")] ((subst_comp_tel RS sym) RSN 
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    (2,trans)) 1);
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by( asm_full_simp_tac (!simpset addsimps [o_def,eq_sym_conv]) 1);
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by (rtac eq_free_eq_subst_tel 1);
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by( safe_tac HOL_cs );
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by (subgoal_tac "ma ~= na" 1);
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by ((forward_tac [new_tv_W] 2) THEN (atac 2));
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   349
by (etac conjE 2);
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   350
by (dtac new_tv_subst_tel 2);
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by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 2);
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by (( forw_inst_tac [("n","m")] (sym RSN (2,new_tv_W)) 2) THEN (atac 2));
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   353
by (etac conjE 2);
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   354
by (dtac (free_tv_app_subst_tel RS subsetD) 2);
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by (fast_tac (set_cs addDs [W_var_geD,new_tv_list_le,codD,
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    new_tv_not_free_tv]) 2);
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   357
by (case_tac "na: free_tv t - free_tv sa" 1);
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   358
(* case na ~: free_tv t - free_tv sa *)
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by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
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(* case na : free_tv t - free_tv sa *)
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by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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by (dtac (free_tv_app_subst_tel RS subsetD) 1);
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by (fast_tac (set_cs addDs [codD,subst_comp_tel RSN (2,trans),
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    eq_subst_tel_eq_free] addss ((!simpset addsimps 
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    [de_Morgan_disj,free_tv_subst,dom_def]))) 1);
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qed_spec_mp "W_complete_lemma";
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   367
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goal W.thy
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 "!!e. [] |- e :: t' ==>  (? s t. (? m. W e [] n = Ok(s,t,m)) &  \
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\                                 (? r. t' = $r t))";
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by(cut_inst_tac [("a","[]"),("s'","id_subst"),("e","e"),("t'","t'")]
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                W_complete_lemma 1);
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by(ALLGOALS Asm_full_simp_tac);
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qed "W_complete";