Simplified proof.
(* Title: HOL/W0/W.ML
ID: $Id$
Author: Dieter Nazareth and Tobias Nipkow
Copyright 1995 TU Muenchen
Correctness and completeness of type inference algorithm W
*)
Addsimps [Suc_le_lessD]; Delsimps [less_imp_le]; (*the combination loops*)
Delsimps (ex_simps @ all_simps);
(* correctness of W with respect to has_type *)
Goal "!a s t m n . Ok (s,t,m) = W e a n --> $s a |- e :: t";
by (induct_tac "e" 1);
(* case Var n *)
by (Asm_simp_tac 1);
(* case Abs e *)
by (asm_full_simp_tac (simpset() addsimps [app_subst_list]
addsplits [split_bind]) 1);
by (strip_tac 1);
by (dtac sym 1);
by (fast_tac (HOL_cs addss simpset()) 1);
(* case App e1 e2 *)
by (simp_tac (simpset() addsplits [split_bind]) 1);
by (strip_tac 1);
by ( rename_tac "sa ta na sb tb nb sc" 1);
by (res_inst_tac [("t2.0","$ sc tb")] has_type.AppI 1);
by (res_inst_tac [("s1","sc")] (app_subst_TVar RS subst) 1);
by (rtac (app_subst_Fun RS subst) 1);
by (res_inst_tac [("t","$sc(tb -> (TVar nb))"),("s","$sc($sb ta)")] subst 1);
by (Asm_full_simp_tac 1);
by (simp_tac (HOL_ss addsimps [subst_comp_tel RS sym]) 1);
by ( (rtac has_type_cl_sub 1) THEN (rtac has_type_cl_sub 1));
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1);
by (asm_full_simp_tac (simpset() addsimps [subst_comp_tel RS sym,o_def,has_type_cl_sub,eq_sym_conv]) 1);
qed_spec_mp "W_correct";
val has_type_casesE = map(has_type.mk_cases expr.simps)
[" s |- Var n :: t"," s |- Abs e :: t","s |- App e1 e2 ::t"];
(* the resulting type variable is always greater or equal than the given one *)
Goal "!a n s t m. W e a n = Ok (s,t,m) --> n<=m";
by (induct_tac "e" 1);
(* case Var(n) *)
by (Clarsimp_tac 1);
(* case Abs e *)
by (simp_tac (simpset() addsplits [split_bind]) 1);
by (fast_tac (HOL_cs addDs [Suc_leD]) 1);
(* case App e1 e2 *)
by (simp_tac (simpset() addsplits [split_bind]) 1);
by (strip_tac 1);
by (rename_tac "s t na sa ta nb sb sc tb m" 1);
by (eres_inst_tac [("x","a")] allE 1);
by (eres_inst_tac [("x","n")] allE 1);
by (eres_inst_tac [("x","$ s a")] allE 1);
by (eres_inst_tac [("x","s")] allE 1);
by (eres_inst_tac [("x","t")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1);
by (etac conjE 1);
by (eres_inst_tac [("x","sa")] allE 1);
by (eres_inst_tac [("x","ta")] allE 1);
by (eres_inst_tac [("x","nb")] allE 1);
by (Asm_full_simp_tac 1);
qed_spec_mp "W_var_ge";
Addsimps [W_var_ge];
Goal "Ok (s,t,m) = W e a n ==> n<=m";
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1);
qed "W_var_geD";
(* auxiliary lemma *)
goal Maybe.thy "(y = Ok x) = (Ok x = y)";
by ( simp_tac (simpset() addsimps [eq_sym_conv]) 1);
qed "rotate_Ok";
(* resulting type variable is new *)
Goal "!n a s t m. new_tv n a --> W e a n = Ok (s,t,m) --> \
\ new_tv m s & new_tv m t";
by (induct_tac "e" 1);
(* case Var n *)
by (Clarsimp_tac 1);
by (force_tac (claset() addEs [list_ball_nth],
simpset() addsimps [id_subst_def,new_tv_list,new_tv_subst])1);
(* case Abs e *)
by (simp_tac (simpset() addsimps [new_tv_subst,new_tv_Suc_list]
addsplits [split_bind]) 1);
by (strip_tac 1);
by (eres_inst_tac [("x","Suc n")] allE 1);
by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
by (fast_tac (HOL_cs addss (simpset()
addsimps [new_tv_subst,new_tv_Suc_list])) 1);
(* case App e1 e2 *)
by (simp_tac (simpset() addsplits [split_bind]) 1);
by (strip_tac 1);
by (rename_tac "s t na sa ta nb sb sc tb m" 1);
by (eres_inst_tac [("x","n")] allE 1);
by (eres_inst_tac [("x","a")] allE 1);
by (eres_inst_tac [("x","s")] allE 1);
by (eres_inst_tac [("x","t")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1);
by (eres_inst_tac [("x","$ s a")] allE 1);
by (eres_inst_tac [("x","sa")] allE 1);
by (eres_inst_tac [("x","ta")] allE 1);
by (eres_inst_tac [("x","nb")] allE 1);
by ( asm_full_simp_tac (simpset() addsimps [o_def,rotate_Ok]) 1);
by (rtac conjI 1);
by (rtac new_tv_subst_comp_2 1);
by (rtac new_tv_subst_comp_2 1);
by (rtac (lessI RS less_imp_le RS new_tv_subst_le) 1);
by (res_inst_tac [("n","na")] new_tv_subst_le 1);
by (asm_full_simp_tac (simpset() addsimps [rotate_Ok]) 1);
by (Asm_simp_tac 1);
by (fast_tac (HOL_cs addDs [W_var_geD] addIs
[new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS new_tv_subst_le])
1);
by (etac (sym RS mgu_new) 1);
by (best_tac (HOL_cs addDs [W_var_geD]
addIs [new_tv_subst_te,new_tv_list_le,
new_tv_subst_tel,
lessI RS less_imp_le RS new_tv_le,
lessI RS less_imp_le RS new_tv_subst_le,
new_tv_le]) 1);
by (fast_tac (HOL_cs addDs [W_var_geD]
addIs [new_tv_list_le,new_tv_subst_tel,new_tv_le]
addss (simpset())) 1);
by (rtac (lessI RS new_tv_subst_var) 1);
by (etac (sym RS mgu_new) 1);
by (best_tac (HOL_cs addSIs [lessI RS less_imp_le RS new_tv_le,new_tv_subst_te]
addDs [W_var_geD]
addIs [new_tv_list_le,
new_tv_subst_tel,
lessI RS less_imp_le RS new_tv_subst_le,
new_tv_le]
addss simpset()) 1);
by (fast_tac (HOL_cs addDs [W_var_geD]
addIs [new_tv_list_le,new_tv_subst_tel,new_tv_le]
addss (simpset())) 1);
qed_spec_mp "new_tv_W";
Goal "!n a s t m v. W e a n = Ok (s,t,m) --> \
\ (v:free_tv s | v:free_tv t) --> v<n --> v:free_tv a";
by (induct_tac "e" 1);
(* case Var n *)
by (Clarsimp_tac 1);
by (fast_tac (HOL_cs addIs [nth_mem,subsetD,ftv_mem_sub_ftv_list])1);
(* case Abs e *)
by (asm_full_simp_tac (simpset() addsimps
[free_tv_subst] addsplits [split_bind]) 1);
by (strip_tac 1);
by (rename_tac "s t na sa ta m v" 1);
by (eres_inst_tac [("x","Suc n")] allE 1);
by (eres_inst_tac [("x","TVar n # a")] allE 1);
by (eres_inst_tac [("x","s")] allE 1);
by (eres_inst_tac [("x","t")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (eres_inst_tac [("x","v")] allE 1);
by (force_tac (claset() addSEs [allE] addIs [cod_app_subst], simpset()) 1);
(** LEVEL 13 **)
(* case App e1 e2 *)
by (simp_tac (simpset() addsplits [split_bind]) 1);
by (strip_tac 1);
by (rename_tac "s t na sa ta nb sb sc tb m v" 1);
by (eres_inst_tac [("x","n")] allE 1);
by (eres_inst_tac [("x","a")] allE 1);
by (eres_inst_tac [("x","s")] allE 1);
by (eres_inst_tac [("x","t")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (eres_inst_tac [("x","na")] allE 1);
by (eres_inst_tac [("x","v")] allE 1);
(** LEVEL 23 **)
(* second case *)
by (eres_inst_tac [("x","$ s a")] allE 1);
by (eres_inst_tac [("x","sa")] allE 1);
by (eres_inst_tac [("x","ta")] allE 1);
by (eres_inst_tac [("x","nb")] allE 1);
by (eres_inst_tac [("x","v")] allE 1);
by (safe_tac (empty_cs addSIs [conjI,impI] addSEs [conjE]) );
by (asm_full_simp_tac (simpset() addsimps [rotate_Ok,o_def]) 1);
by (dtac W_var_geD 1);
by (dtac W_var_geD 1);
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
(** LEVEL 33 **)
by (fast_tac (HOL_cs addDs [free_tv_comp_subst RS subsetD,sym RS mgu_free,
codD,free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD,
subsetD]
addEs [UnE]
addss simpset()) 1);
by (Asm_full_simp_tac 1);
by (dtac (sym RS W_var_geD) 1);
by (dtac (sym RS W_var_geD) 1);
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
by (fast_tac (HOL_cs addDs [mgu_free, codD,free_tv_subst_var RS subsetD,
free_tv_app_subst_te RS subsetD,
free_tv_app_subst_tel RS subsetD,
less_le_trans,subsetD]
addSEs [UnE]
addss (simpset() setSolver unsafe_solver)) 1);
(* builtin arithmetic in simpset messes things up *)
qed_spec_mp "free_tv_W";
(* Completeness of W w.r.t. has_type *)
Goal "!s' a t' n. $s' a |- e :: t' --> new_tv n a --> \
\ (? s t. (? m. W e a n = Ok (s,t,m)) & \
\ (? r. $s' a = $r ($s a) & t' = $r t))";
by (induct_tac "e" 1);
(* case Var n *)
by (strip_tac 1);
by (simp_tac (simpset() addcongs [conj_cong]) 1);
by (eresolve_tac has_type_casesE 1);
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv,app_subst_list]) 1);
by (res_inst_tac [("x","s'")] exI 1);
by (Asm_simp_tac 1);
(** LEVEL 10 **)
(* case Abs e *)
by (strip_tac 1);
by (eresolve_tac has_type_casesE 1);
by (eres_inst_tac [("x","%x. if x=n then t1 else (s' x)")] allE 1);
by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
by (eres_inst_tac [("x","t2")] allE 1);
by (eres_inst_tac [("x","Suc n")] allE 1);
by (fast_tac (HOL_cs addss (simpset() addcongs [conj_cong]
addsplits [split_bind])) 1);
(** LEVEL 17 **)
(* case App e1 e2 *)
by (strip_tac 1);
by (eresolve_tac has_type_casesE 1);
by (eres_inst_tac [("x","s'")] allE 1);
by (eres_inst_tac [("x","a")] allE 1);
by (eres_inst_tac [("x","t2 -> t'")] allE 1);
by (eres_inst_tac [("x","n")] allE 1);
by (safe_tac HOL_cs);
by (eres_inst_tac [("x","r")] allE 1);
by (eres_inst_tac [("x","$ s a")] allE 1);
by (eres_inst_tac [("x","t2")] allE 1);
by (eres_inst_tac [("x","m")] allE 1);
by (Asm_full_simp_tac 1);
by (safe_tac HOL_cs);
by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS
conjunct1,new_tv_list_le,new_tv_subst_tel]) 1);
(** LEVEL 35 **)
by (subgoal_tac
"$ (%x. if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
\ else ra x)) ($ sa t) = \
\ $ (%x. if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
\ else ra x)) (ta -> (TVar ma))" 1);
by (res_inst_tac [("t","$ (%x. if x = ma then t' else \
\ (if x:(free_tv t - free_tv sa) then r x else ra x)) ($ sa t)"),
("s","($ ra ta) -> t'")] ssubst 2);
by (asm_simp_tac (simpset() addsimps [subst_comp_te]) 2);
by (rtac eq_free_eq_subst_te 2);
by (strip_tac 2);
by (subgoal_tac "na ~=ma" 2);
by (fast_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD,
new_tv_not_free_tv,new_tv_le]) 3);
(** LEVEL 42 **)
by (case_tac "na:free_tv sa" 2);
(* na ~: free_tv sa *)
by (forward_tac [not_free_impl_id] 3);
by (Asm_simp_tac 3);
(* na : free_tv sa *)
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
by (dtac eq_subst_tel_eq_free 2);
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
by (Asm_simp_tac 2);
by (case_tac "na:dom sa" 2);
(* na ~: dom sa *)
(** LEVEL 50 **)
by (asm_full_simp_tac (simpset() addsimps [dom_def]) 3);
(* na : dom sa *)
by (rtac eq_free_eq_subst_te 2);
by (strip_tac 2);
by (subgoal_tac "nb ~= ma" 2);
by ((forward_tac [new_tv_W] 3) THEN (atac 3));
by (etac conjE 3);
by (dtac new_tv_subst_tel 3);
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 3);
by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss
(simpset() addsimps [cod_def,free_tv_subst])) 3);
by (fast_tac (set_cs addss (simpset() addsimps [cod_def,free_tv_subst])) 2);
(** LEVEL 60 **)
by (Simp_tac 2);
by (rtac eq_free_eq_subst_te 2);
by (strip_tac 2 );
by (subgoal_tac "na ~= ma" 2);
by ((forward_tac [new_tv_W] 3) THEN (atac 3));
by (etac conjE 3);
by (dtac (sym RS W_var_geD) 3);
by (fast_tac (HOL_cs addDs [new_tv_list_le,new_tv_subst_tel,new_tv_W,new_tv_not_free_tv]) 3);
by (case_tac "na: free_tv t - free_tv sa" 2);
(** LEVEL 69 **)
(* case na ~: free_tv t - free_tv sa *)
by (Asm_full_simp_tac 3);
(* case na : free_tv t - free_tv sa *)
by (Asm_full_simp_tac 2);
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
by (dtac eq_subst_tel_eq_free 2);
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
by (asm_full_simp_tac (simpset() addsimps [free_tv_subst,dom_def]) 2);
by (Fast_tac 2);
(** LEVEL 76 **)
by (asm_simp_tac (simpset() addsplits [split_bind]) 1);
by (safe_tac HOL_cs);
by (dtac mgu_Ok 1);
by ( fast_tac (HOL_cs addss simpset()) 1);
by ((dtac mgu_mg 1) THEN (atac 1));
by (etac exE 1);
by (res_inst_tac [("x","rb")] exI 1);
by (rtac conjI 1);
by (dres_inst_tac [("x","ma")] fun_cong 2);
by ( asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 2);
by (simp_tac (simpset() addsimps [o_def,subst_comp_tel RS sym]) 1);
(** LEVEL 90 **)
by (rtac ((subst_comp_tel RS sym) RSN (2,trans)) 1);
by ( asm_full_simp_tac (simpset() addsimps [o_def,eq_sym_conv]) 1);
by (rtac eq_free_eq_subst_tel 1);
by ( safe_tac HOL_cs );
by (subgoal_tac "ma ~= na" 1);
by ((forward_tac [new_tv_W] 2) THEN (atac 2));
by (etac conjE 2);
by (dtac new_tv_subst_tel 2);
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 2);
by (forw_inst_tac [("n","m")] new_tv_W 2 THEN atac 2);
(** LEVEL 100 **)
by (etac conjE 2);
by (dtac (free_tv_app_subst_tel RS subsetD) 2);
by (fast_tac (set_cs addDs [sym RS W_var_geD,new_tv_list_le,codD,
new_tv_not_free_tv]) 2);
by (case_tac "na: free_tv t - free_tv sa" 1);
(* case na ~: free_tv t - free_tv sa *)
by (Asm_full_simp_tac 2);
(* case na : free_tv t - free_tv sa *)
by (Asm_full_simp_tac 1);
by (dtac (free_tv_app_subst_tel RS subsetD) 1);
by (fast_tac (set_cs addDs [codD,subst_comp_tel RSN (2,trans),
eq_subst_tel_eq_free]
addss ((simpset() addsimps [free_tv_subst,dom_def]))) 1);
(** LEVEL 106 **)
by (Fast_tac 1);
qed_spec_mp "W_complete_lemma";
Goal "[] |- e :: t' ==> (? s t. (? m. W e [] n = Ok(s,t,m)) & \
\ (? r. t' = $r t))";
by (cut_inst_tac [("a","[]"),("s'","id_subst"),("e","e"),("t'","t'")]
W_complete_lemma 1);
by (ALLGOALS Asm_full_simp_tac);
qed "W_complete";