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(* Title: HOL/MiniML/W.thy
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ID: $Id$
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Author: Dieter Nazareth, Wolfgang Naraschewski and Tobias Nipkow
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Copyright 1996 TU Muenchen
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Correctness and completeness of type inference algorithm W
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*)
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theory W = MiniML:
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types result_W = "(subst * typ * nat)option"
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-- "type inference algorithm W"
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consts W :: "[expr, ctxt, nat] => result_W"
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primrec
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"W (Var i) A n =
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(if i < length A then Some( id_subst,
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bound_typ_inst (%b. TVar(b+n)) (A!i),
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n + (min_new_bound_tv (A!i)) )
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else None)"
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"W (Abs e) A n = ( (S,t,m) := W e ((FVar n)#A) (Suc n);
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Some( S, (S n) -> t, m) )"
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"W (App e1 e2) A n = ( (S1,t1,m1) := W e1 A n;
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(S2,t2,m2) := W e2 ($S1 A) m1;
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U := mgu ($S2 t1) (t2 -> (TVar m2));
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Some( $U o $S2 o S1, U m2, Suc m2) )"
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"W (LET e1 e2) A n = ( (S1,t1,m1) := W e1 A n;
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(S2,t2,m2) := W e2 ((gen ($S1 A) t1)#($S1 A)) m1;
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Some( $S2 o S1, t2, m2) )"
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declare Suc_le_lessD [simp]
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declare less_imp_le [simp del] (*the combination loops*)
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inductive_cases has_type_casesE:
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"A |- Var n :: t"
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"A |- Abs e :: t"
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"A |- App e1 e2 ::t"
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"A |- LET e1 e2 ::t"
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(* the resulting type variable is always greater or equal than the given one *)
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lemma W_var_ge [rule_format (no_asm)]: "!A n S t m. W e A n = Some (S,t,m) --> n<=m"
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apply (induct_tac "e")
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(* case Var(n) *)
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apply (simp (no_asm) split add: split_option_bind)
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(* case Abs e *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (fast dest: Suc_leD)
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(* case App e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (blast intro: le_SucI le_trans)
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(* case LET e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (blast intro: le_trans)
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done
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declare W_var_ge [simp]
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lemma W_var_geD: "Some (S,t,m) = W e A n ==> n<=m"
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apply (simp add: eq_sym_conv)
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done
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lemma new_tv_compatible_W: "new_tv n A ==> Some (S,t,m) = W e A n ==> new_tv m A"
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apply (drule W_var_geD)
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apply (rule new_scheme_list_le)
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apply assumption
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apply assumption
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done
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lemma new_tv_bound_typ_inst_sch [rule_format (no_asm)]: "new_tv n sch --> new_tv (n + (min_new_bound_tv sch)) (bound_typ_inst (%b. TVar (b + n)) sch)"
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apply (induct_tac "sch")
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apply simp
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apply (simp add: add_commute)
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apply (intro strip)
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apply simp
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apply (erule conjE)
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apply (rule conjI)
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apply (rule new_tv_le)
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prefer 2 apply (assumption)
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apply (rule add_le_mono)
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apply (simp (no_asm))
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apply (simp (no_asm) add: max_def)
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apply (rule new_tv_le)
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prefer 2 apply (assumption)
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apply (rule add_le_mono)
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apply (simp (no_asm))
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apply (simp (no_asm) add: max_def)
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done
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declare new_tv_bound_typ_inst_sch [simp]
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(* resulting type variable is new *)
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lemma new_tv_W [rule_format (no_asm)]: "!n A S t m. new_tv n A --> W e A n = Some (S,t,m) -->
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new_tv m S & new_tv m t"
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apply (induct_tac "e")
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(* case Var n *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (intro strip)
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apply (drule new_tv_nth_nat_A)
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apply assumption
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apply (simp (no_asm_simp))
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(* case Abs e *)
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apply (simp (no_asm) add: new_tv_subst new_tv_Suc_list split add: split_option_bind)
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apply (intro strip)
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apply (erule_tac x = "Suc n" in allE)
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apply (erule_tac x = " (FVar n) #A" in allE)
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apply (fastsimp simp add: new_tv_subst new_tv_Suc_list)
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(* case App e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (intro strip)
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apply (rename_tac S1 t1 n1 S2 t2 n2 S3)
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apply (erule_tac x = "n" in allE)
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apply (erule_tac x = "A" in allE)
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apply (erule_tac x = "S1" in allE)
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apply (erule_tac x = "t1" in allE)
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apply (erule_tac x = "n1" in allE)
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apply (erule_tac x = "n1" in allE)
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apply (simp add: eq_sym_conv del: all_simps)
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apply (erule_tac x = "$S1 A" in allE)
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apply (erule_tac x = "S2" in allE)
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apply (erule_tac x = "t2" in allE)
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apply (erule_tac x = "n2" in allE)
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apply ( simp add: o_def rotate_Some)
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apply (rule conjI)
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apply (rule new_tv_subst_comp_2)
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apply (rule new_tv_subst_comp_2)
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apply (rule lessI [THEN less_imp_le, THEN new_tv_le])
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apply (rule_tac n = "n1" in new_tv_subst_le)
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apply (simp add: rotate_Some)
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apply (simp (no_asm_simp))
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apply (fast dest: W_var_geD intro: new_scheme_list_le new_tv_subst_scheme_list lessI [THEN less_imp_le [THEN new_tv_subst_le]])
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apply (erule sym [THEN mgu_new])
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apply (blast dest!: W_var_geD
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intro: lessI [THEN less_imp_le, THEN new_tv_le] new_tv_subst_te
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new_tv_subst_scheme_list new_scheme_list_le new_tv_le)
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apply (erule impE)
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apply (blast dest: W_var_geD intro: new_tv_subst_scheme_list new_scheme_list_le new_tv_le)
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apply clarsimp
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apply (rule lessI [THEN new_tv_subst_var])
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apply (erule sym [THEN mgu_new])
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apply (blast dest!: W_var_geD
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intro: lessI [THEN less_imp_le, THEN new_tv_le] new_tv_subst_te
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new_tv_subst_scheme_list new_scheme_list_le new_tv_le)
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apply (erule impE)
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apply (blast dest: W_var_geD intro: new_tv_subst_scheme_list new_scheme_list_le new_tv_le)
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apply clarsimp
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-- "41: case LET e1 e2"
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apply (simp (no_asm) split add: split_option_bind)
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apply (intro strip)
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apply (erule allE,erule allE,erule allE,erule allE,erule allE, erule impE, assumption, erule impE, assumption)
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apply (erule conjE)
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apply (erule allE,erule allE,erule allE,erule allE,erule allE, erule impE, erule_tac [2] notE impE, tactic "assume_tac 2")
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apply (simp only: new_tv_def)
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apply (simp (no_asm_simp))
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apply (drule W_var_ge)+
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apply (rule allI)
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apply (intro strip)
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apply (simp only: free_tv_subst)
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apply (drule free_tv_app_subst_scheme_list [THEN subsetD])
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apply (best elim: less_le_trans)
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apply (erule conjE)
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apply (rule conjI)
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apply (simp only: o_def)
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apply (rule new_tv_subst_comp_2)
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apply (erule W_var_ge [THEN new_tv_subst_le])
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apply assumption
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apply assumption
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apply assumption
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done
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lemma free_tv_bound_typ_inst1 [rule_format (no_asm)]: "(v ~: free_tv sch) --> (v : free_tv (bound_typ_inst (TVar o S) sch)) --> (? x. v = S x)"
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apply (induct_tac "sch")
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apply simp
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apply simp
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apply (intro strip)
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apply (rule exI)
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apply (rule refl)
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apply simp
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done
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declare free_tv_bound_typ_inst1 [simp]
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lemma free_tv_W [rule_format (no_asm)]: "!n A S t m v. W e A n = Some (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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apply (induct_tac "e")
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(* case Var n *)
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apply (simp (no_asm) add: free_tv_subst split add: split_option_bind)
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apply (intro strip)
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apply (case_tac "v : free_tv (A!nat) ")
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apply simp
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apply (drule free_tv_bound_typ_inst1)
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apply (simp (no_asm) add: o_def)
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apply (erule exE)
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apply simp
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(* case Abs e *)
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apply (simp add: free_tv_subst split add: split_option_bind del: all_simps)
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apply (intro strip)
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apply (rename_tac S t n1 v)
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apply (erule_tac x = "Suc n" in allE)
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apply (erule_tac x = "FVar n # A" in allE)
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apply (erule_tac x = "S" in allE)
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apply (erule_tac x = "t" in allE)
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apply (erule_tac x = "n1" in allE)
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apply (erule_tac x = "v" in allE)
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apply (bestsimp intro: cod_app_subst simp add: less_Suc_eq)
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(* case App e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind del: all_simps)
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apply (intro strip)
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apply (rename_tac S t n1 S1 t1 n2 S2 v)
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apply (erule_tac x = "n" in allE)
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apply (erule_tac x = "A" in allE)
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apply (erule_tac x = "S" in allE)
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apply (erule_tac x = "t" in allE)
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apply (erule_tac x = "n1" in allE)
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apply (erule_tac x = "n1" in allE)
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apply (erule_tac x = "v" in allE)
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(* second case *)
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apply (erule_tac x = "$ S A" in allE)
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apply (erule_tac x = "S1" in allE)
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apply (erule_tac x = "t1" in allE)
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apply (erule_tac x = "n2" in allE)
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apply (erule_tac x = "v" in allE)
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apply (intro conjI impI | elim conjE)+
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apply (simp add: rotate_Some o_def)
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apply (drule W_var_geD)
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apply (drule W_var_geD)
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apply ( (frule less_le_trans) , (assumption))
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apply clarsimp
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apply (drule free_tv_comp_subst [THEN subsetD])
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apply (drule sym [THEN mgu_free])
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apply clarsimp
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apply (fastsimp dest: free_tv_comp_subst [THEN subsetD] sym [THEN mgu_free] codD free_tv_app_subst_te [THEN subsetD] free_tv_app_subst_scheme_list [THEN subsetD] less_le_trans less_not_refl2 subsetD)
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apply simp
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apply (drule sym [THEN W_var_geD])
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apply (drule sym [THEN W_var_geD])
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apply ( (frule less_le_trans) , (assumption))
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apply clarsimp
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apply (drule mgu_free)
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apply (fastsimp dest: mgu_free codD free_tv_subst_var [THEN subsetD] free_tv_app_subst_te [THEN subsetD] free_tv_app_subst_scheme_list [THEN subsetD] less_le_trans subsetD)
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(* LET e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind del: all_simps)
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apply (intro strip)
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apply (rename_tac S1 t1 n2 S2 t2 n3 v)
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apply (erule_tac x = "n" in allE)
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apply (erule_tac x = "A" in allE)
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apply (erule_tac x = "S1" in allE)
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apply (erule_tac x = "t1" in allE)
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apply (rotate_tac -1)
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apply (erule_tac x = "n2" in allE)
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apply (rotate_tac -1)
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apply (erule_tac x = "v" in allE)
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apply (erule (1) notE impE)
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apply (erule allE,erule allE,erule allE,erule allE,erule allE,erule_tac x = "v" in allE)
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apply (erule (1) notE impE)
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apply simp
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apply (rule conjI)
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apply (fast dest!: codD free_tv_app_subst_scheme_list [THEN subsetD] free_tv_o_subst [THEN subsetD] W_var_ge dest: less_le_trans)
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apply (fast dest!: codD free_tv_app_subst_scheme_list [THEN subsetD] W_var_ge dest: less_le_trans)
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done
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lemma weaken_A_Int_B_eq_empty: "(!x. x : A --> x ~: B) ==> A Int B = {}"
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apply fast
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done
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lemma weaken_not_elem_A_minus_B: "x ~: A | x : B ==> x ~: A - B"
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apply fast
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done
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(* correctness of W with respect to has_type *)
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lemma W_correct_lemma [rule_format (no_asm)]: "!A S t m n . new_tv n A --> Some (S,t,m) = W e A n --> $S A |- e :: t"
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apply (induct_tac "e")
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(* case Var n *)
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apply simp
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apply (intro strip)
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apply (rule has_type.VarI)
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apply (simp (no_asm))
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apply (simp (no_asm) add: is_bound_typ_instance)
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apply (rule exI)
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apply (rule refl)
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(* case Abs e *)
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apply (simp add: app_subst_list split add: split_option_bind)
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apply (intro strip)
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apply (erule_tac x = " (mk_scheme (TVar n)) # A" in allE)
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apply simp
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apply (rule has_type.AbsI)
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apply (drule le_refl [THEN le_SucI, THEN new_scheme_list_le])
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apply (drule sym)
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apply (erule allE)+
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apply (erule impE)
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apply (erule_tac [2] notE impE, tactic "assume_tac 2")
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apply (simp (no_asm_simp))
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apply assumption
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(* case App e1 e2 *)
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apply (simp (no_asm) split add: split_option_bind)
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apply (intro strip)
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apply (rename_tac S1 t1 n1 S2 t2 n2 S3)
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apply (rule_tac ?t2.0 = "$ S3 t2" in has_type.AppI)
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apply (rule_tac S1 = "S3" in app_subst_TVar [THEN subst])
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apply (rule app_subst_Fun [THEN subst])
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apply (rule_tac t = "$S3 (t2 -> (TVar n2))" and s = "$S3 ($S2 t1) " in subst)
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apply simp
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apply (simp only: subst_comp_scheme_list [symmetric] o_def)
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apply ((rule has_type_cl_sub [THEN spec]) , (rule has_type_cl_sub [THEN spec]))
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apply (simp add: eq_sym_conv)
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apply (simp add: subst_comp_scheme_list [symmetric] o_def has_type_cl_sub eq_sym_conv)
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apply (rule has_type_cl_sub [THEN spec])
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apply (frule new_tv_W)
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apply assumption
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apply (drule conjunct1)
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apply (frule new_tv_subst_scheme_list)
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apply (rule new_scheme_list_le)
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apply (rule W_var_ge)
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apply assumption
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325 |
apply assumption
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326 |
apply (erule thin_rl)
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327 |
apply (erule allE)+
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328 |
apply (drule sym)
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329 |
apply (drule sym)
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330 |
apply (erule thin_rl)
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331 |
apply (erule thin_rl)
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332 |
apply (erule (1) notE impE)
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333 |
apply (erule (1) notE impE)
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334 |
apply assumption
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335 |
(* case Let e1 e2 *)
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336 |
apply (simp (no_asm) split add: split_option_bind)
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337 |
apply (intro strip)
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338 |
(*by (rename_tac "w q m1 S1 t1 m2 S2 t n2" 1); *)
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339 |
apply (rename_tac S1 t1 m1 S2)
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340 |
apply (rule_tac ?t1.0 = "$ S2 t1" in has_type.LETI)
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341 |
apply (simp (no_asm) add: o_def)
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342 |
apply (simp only: subst_comp_scheme_list [symmetric])
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343 |
apply (rule has_type_cl_sub [THEN spec])
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344 |
apply (drule_tac x = "A" in spec)
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345 |
apply (drule_tac x = "S1" in spec)
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346 |
apply (drule_tac x = "t1" in spec)
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347 |
apply (drule_tac x = "m1" in spec)
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348 |
apply (drule_tac x = "n" in spec)
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349 |
apply (erule (1) notE impE)
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350 |
apply (simp add: eq_sym_conv)
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351 |
apply (simp (no_asm) add: o_def)
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|
352 |
apply (simp only: subst_comp_scheme_list [symmetric])
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353 |
apply (rule gen_subst_commutes [symmetric, THEN subst])
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354 |
apply (rule_tac [2] app_subst_Cons [THEN subst])
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355 |
apply (erule_tac [2] thin_rl)
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356 |
apply (drule_tac [2] x = "gen ($S1 A) t1 # $ S1 A" in spec)
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357 |
apply (drule_tac [2] x = "S2" in spec)
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358 |
apply (drule_tac [2] x = "t" in spec)
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359 |
apply (drule_tac [2] x = "m" in spec)
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360 |
apply (drule_tac [2] x = "m1" in spec)
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|
361 |
apply (frule_tac [2] new_tv_W)
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362 |
prefer 2 apply (assumption)
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363 |
apply (drule_tac [2] conjunct1)
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364 |
apply (frule_tac [2] new_tv_subst_scheme_list)
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365 |
apply (rule_tac [2] new_scheme_list_le)
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366 |
apply (rule_tac [2] W_var_ge)
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367 |
prefer 2 apply (assumption)
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368 |
prefer 2 apply (assumption)
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369 |
apply (erule_tac [2] impE)
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|
370 |
apply (rule_tac [2] A = "$ S1 A" in new_tv_only_depends_on_free_tv_scheme_list)
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371 |
prefer 2 apply simp
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372 |
apply (fast)
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|
373 |
prefer 2 apply (assumption)
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374 |
prefer 2 apply simp
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|
375 |
apply (rule weaken_A_Int_B_eq_empty)
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|
376 |
apply (rule allI)
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|
377 |
apply (intro strip)
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|
378 |
apply (rule weaken_not_elem_A_minus_B)
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|
379 |
apply (case_tac "x < m1")
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|
380 |
apply (rule disjI2)
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|
381 |
apply (rule free_tv_gen_cons [THEN subst])
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|
382 |
apply (rule free_tv_W)
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|
383 |
apply assumption
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|
384 |
apply simp
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|
385 |
apply assumption
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|
386 |
apply (rule disjI1)
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|
387 |
apply (drule new_tv_W)
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|
388 |
apply assumption
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|
389 |
apply (drule conjunct2)
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|
390 |
apply (rule new_tv_not_free_tv)
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|
391 |
apply (rule new_tv_le)
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|
392 |
prefer 2 apply (assumption)
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|
393 |
apply (simp add: not_less_iff_le)
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|
394 |
done
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|
395 |
|
|
396 |
(* Completeness of W w.r.t. has_type *)
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|
397 |
lemma W_complete_lemma [rule_format (no_asm)]:
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|
398 |
"ALL S' A t' n. $S' A |- e :: t' --> new_tv n A -->
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|
399 |
(EX S t. (EX m. W e A n = Some (S,t,m)) &
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|
400 |
(EX R. $S' A = $R ($S A) & t' = $R t))"
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|
401 |
apply (induct_tac "e")
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|
402 |
(* case Var n *)
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|
403 |
apply (intro strip)
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|
404 |
apply (simp (no_asm) cong add: conj_cong)
|
|
405 |
apply (erule has_type_casesE)
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|
406 |
apply (simp add: is_bound_typ_instance)
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|
407 |
apply (erule exE)
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|
408 |
apply (hypsubst)
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|
409 |
apply (rename_tac "S")
|
|
410 |
apply (rule_tac x = "%x. (if x < n then S' x else S (x - n))" in exI)
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|
411 |
apply (rule conjI)
|
|
412 |
apply (simp (no_asm_simp))
|
|
413 |
apply (simp (no_asm_simp) add: bound_typ_inst_composed_subst [symmetric] new_tv_nth_nat_A o_def nth_subst
|
|
414 |
del: bound_typ_inst_composed_subst)
|
|
415 |
|
|
416 |
(* case Abs e *)
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|
417 |
apply (intro strip)
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|
418 |
apply (erule has_type_casesE)
|
|
419 |
apply (erule_tac x = "%x. if x=n then t1 else (S' x) " in allE)
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|
420 |
apply (erule_tac x = " (FVar n) #A" in allE)
|
|
421 |
apply (erule_tac x = "t2" in allE)
|
|
422 |
apply (erule_tac x = "Suc n" in allE)
|
|
423 |
apply (bestsimp dest!: mk_scheme_injective cong: conj_cong split: split_option_bind)
|
|
424 |
|
|
425 |
(* case App e1 e2 *)
|
|
426 |
apply (intro strip)
|
|
427 |
apply (erule has_type_casesE)
|
|
428 |
apply (erule_tac x = "S'" in allE)
|
|
429 |
apply (erule_tac x = "A" in allE)
|
|
430 |
apply (erule_tac x = "t2 -> t'" in allE)
|
|
431 |
apply (erule_tac x = "n" in allE)
|
|
432 |
apply safe
|
|
433 |
apply (erule_tac x = "R" in allE)
|
|
434 |
apply (erule_tac x = "$ S A" in allE)
|
|
435 |
apply (erule_tac x = "t2" in allE)
|
|
436 |
apply (erule_tac x = "m" in allE)
|
|
437 |
apply simp
|
|
438 |
apply safe
|
|
439 |
apply (blast intro: sym [THEN W_var_geD] new_tv_W [THEN conjunct1] new_scheme_list_le new_tv_subst_scheme_list)
|
|
440 |
|
|
441 |
(** LEVEL 33 **)
|
|
442 |
apply (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))")
|
|
443 |
apply (rule_tac [2] t = "$ (%x. if x = ma then t' else (if x: (free_tv t - free_tv Sa) then R x else Ra x)) ($ Sa t) " and s = " ($ Ra ta) -> t'" in ssubst)
|
|
444 |
prefer 2 apply (simp (no_asm_simp) add: subst_comp_te) prefer 2
|
|
445 |
apply (rule_tac [2] eq_free_eq_subst_te)
|
|
446 |
prefer 2 apply (intro strip) prefer 2
|
|
447 |
apply (subgoal_tac [2] "na ~=ma")
|
|
448 |
prefer 3 apply (best dest: new_tv_W sym [THEN W_var_geD] new_tv_not_free_tv new_tv_le)
|
|
449 |
apply (case_tac [2] "na:free_tv Sa")
|
|
450 |
(* n1 ~: free_tv S1 *)
|
|
451 |
apply (frule_tac [3] not_free_impl_id)
|
|
452 |
prefer 3 apply (simp)
|
|
453 |
(* na : free_tv Sa *)
|
|
454 |
apply (drule_tac [2] A1 = "$ S A" in trans [OF _ subst_comp_scheme_list])
|
|
455 |
apply (drule_tac [2] eq_subst_scheme_list_eq_free)
|
|
456 |
prefer 2 apply (fast intro: free_tv_W free_tv_le_new_tv dest: new_tv_W)
|
|
457 |
prefer 2 apply (simp (no_asm_simp)) prefer 2
|
|
458 |
apply (case_tac [2] "na:dom Sa")
|
|
459 |
(* na ~: dom Sa *)
|
|
460 |
prefer 3 apply (simp add: dom_def)
|
|
461 |
(* na : dom Sa *)
|
|
462 |
apply (rule_tac [2] eq_free_eq_subst_te)
|
|
463 |
prefer 2 apply (intro strip) prefer 2
|
|
464 |
apply (subgoal_tac [2] "nb ~= ma")
|
|
465 |
apply (frule_tac [3] new_tv_W) prefer 3 apply assumption
|
|
466 |
apply (erule_tac [3] conjE)
|
|
467 |
apply (drule_tac [3] new_tv_subst_scheme_list)
|
|
468 |
prefer 3 apply (fast intro: new_scheme_list_le dest: sym [THEN W_var_geD])
|
|
469 |
prefer 3 apply (fastsimp dest: new_tv_W new_tv_not_free_tv simp add: cod_def free_tv_subst)
|
|
470 |
prefer 2 apply (fastsimp simp add: cod_def free_tv_subst)
|
|
471 |
prefer 2 apply (simp (no_asm)) prefer 2
|
|
472 |
apply (rule_tac [2] eq_free_eq_subst_te)
|
|
473 |
prefer 2 apply (intro strip) prefer 2
|
|
474 |
apply (subgoal_tac [2] "na ~= ma")
|
|
475 |
apply (frule_tac [3] new_tv_W) prefer 3 apply assumption
|
|
476 |
apply (erule_tac [3] conjE)
|
|
477 |
apply (drule_tac [3] sym [THEN W_var_geD])
|
|
478 |
prefer 3 apply (fast dest: new_scheme_list_le new_tv_subst_scheme_list new_tv_W new_tv_not_free_tv)
|
|
479 |
apply (case_tac [2] "na: free_tv t - free_tv Sa")
|
|
480 |
(* case na ~: free_tv t - free_tv Sa *)
|
|
481 |
prefer 3
|
|
482 |
apply simp
|
|
483 |
apply fast
|
|
484 |
(* case na : free_tv t - free_tv Sa *)
|
|
485 |
prefer 2 apply simp prefer 2
|
|
486 |
apply (drule_tac [2] A1 = "$ S A" and r = "$ R ($ S A)" in trans [OF _ subst_comp_scheme_list])
|
|
487 |
apply (drule_tac [2] eq_subst_scheme_list_eq_free)
|
|
488 |
prefer 2
|
|
489 |
apply (fast intro: free_tv_W free_tv_le_new_tv dest: new_tv_W)
|
|
490 |
(** LEVEL 73 **)
|
|
491 |
prefer 2 apply (simp add: free_tv_subst dom_def)
|
|
492 |
apply (simp (no_asm_simp) split add: split_option_bind)
|
|
493 |
apply safe
|
|
494 |
apply (drule mgu_Some)
|
|
495 |
apply fastsimp
|
|
496 |
(** LEVEL 78 *)
|
|
497 |
apply (drule mgu_mg, assumption)
|
|
498 |
apply (erule exE)
|
|
499 |
apply (rule_tac x = "Rb" in exI)
|
|
500 |
apply (rule conjI)
|
|
501 |
apply (drule_tac [2] x = "ma" in fun_cong)
|
|
502 |
prefer 2 apply (simp add: eq_sym_conv)
|
|
503 |
apply (simp (no_asm) add: subst_comp_scheme_list)
|
|
504 |
apply (simp (no_asm) add: subst_comp_scheme_list [symmetric])
|
|
505 |
apply (rule_tac A2 = "($ Sa ($ S A))" in trans [OF _ subst_comp_scheme_list [symmetric]])
|
|
506 |
apply (simp add: o_def eq_sym_conv)
|
|
507 |
apply (drule_tac s = "Some ?X" in sym)
|
|
508 |
apply (rule eq_free_eq_subst_scheme_list)
|
|
509 |
apply safe
|
|
510 |
apply (subgoal_tac "ma ~= na")
|
|
511 |
apply (frule_tac [2] new_tv_W) prefer 2 apply assumption
|
|
512 |
apply (erule_tac [2] conjE)
|
|
513 |
apply (drule_tac [2] new_tv_subst_scheme_list)
|
|
514 |
prefer 2 apply (fast intro: new_scheme_list_le dest: sym [THEN W_var_geD])
|
|
515 |
apply (frule_tac [2] n = "m" in new_tv_W) prefer 2 apply assumption
|
|
516 |
apply (erule_tac [2] conjE)
|
|
517 |
apply (drule_tac [2] free_tv_app_subst_scheme_list [THEN subsetD])
|
|
518 |
apply (tactic {*
|
|
519 |
(fast_tac (set_cs addDs [sym RS thm "W_var_geD", thm "new_scheme_list_le", thm "codD",
|
|
520 |
thm "new_tv_not_free_tv"]) 2) *})
|
|
521 |
apply (case_tac "na: free_tv t - free_tv Sa")
|
|
522 |
(* case na ~: free_tv t - free_tv Sa *)
|
|
523 |
prefer 2 apply simp apply blast
|
|
524 |
(* case na : free_tv t - free_tv Sa *)
|
|
525 |
apply simp
|
|
526 |
apply (drule free_tv_app_subst_scheme_list [THEN subsetD])
|
|
527 |
apply (fastsimp dest: codD trans [OF _ subst_comp_scheme_list]
|
|
528 |
eq_subst_scheme_list_eq_free
|
|
529 |
simp add: free_tv_subst dom_def)
|
|
530 |
(* case Let e1 e2 *)
|
|
531 |
apply (erule has_type_casesE)
|
|
532 |
apply (erule_tac x = "S'" in allE)
|
|
533 |
apply (erule_tac x = "A" in allE)
|
|
534 |
apply (erule_tac x = "t1" in allE)
|
|
535 |
apply (erule_tac x = "n" in allE)
|
|
536 |
apply (erule (1) notE impE)
|
|
537 |
apply (erule (1) notE impE)
|
|
538 |
apply safe
|
|
539 |
apply (simp (no_asm_simp))
|
|
540 |
apply (erule_tac x = "R" in allE)
|
|
541 |
apply (erule_tac x = "gen ($ S A) t # $ S A" in allE)
|
|
542 |
apply (erule_tac x = "t'" in allE)
|
|
543 |
apply (erule_tac x = "m" in allE)
|
|
544 |
apply simp
|
|
545 |
apply (drule mp)
|
|
546 |
apply (rule has_type_le_env)
|
|
547 |
apply assumption
|
|
548 |
apply (simp (no_asm))
|
|
549 |
apply (rule gen_bound_typ_instance)
|
|
550 |
apply (drule mp)
|
|
551 |
apply (frule new_tv_compatible_W)
|
|
552 |
apply (rule sym)
|
|
553 |
apply assumption
|
|
554 |
apply (fast dest: new_tv_compatible_gen new_tv_subst_scheme_list new_tv_W)
|
|
555 |
apply safe
|
|
556 |
apply simp
|
|
557 |
apply (rule_tac x = "Ra" in exI)
|
|
558 |
apply (simp (no_asm) add: o_def subst_comp_scheme_list [symmetric])
|
|
559 |
done
|
|
560 |
|
|
561 |
|
|
562 |
lemma W_complete: "[] |- e :: t' ==> (? S t. (? m. W e [] n = Some(S,t,m)) &
|
|
563 |
(? R. t' = $ R t))"
|
|
564 |
apply (cut_tac A = "[]" and S' = "id_subst" and e = "e" and t' = "t'" in W_complete_lemma)
|
|
565 |
apply simp_all
|
|
566 |
done
|
|
567 |
|
1300
|
568 |
end
|