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(* Title: HOL/Lambda/InductTermi.ML
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1998 TU Muenchen
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*)
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(*** Every term in IT terminates ***)
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Goal "s : termi beta ==> !t. t : termi beta --> \
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\ (!r ss. t = r[s/0]$$ss --> Abs r $ s $$ ss : termi beta)";
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by (etac acc_induct 1);
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by (etac thin_rl 1);
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by (rtac allI 1);
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by (rtac impI 1);
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by (etac acc_induct 1);
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by (Clarify_tac 1);
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by (rtac accI 1);
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by (safe_tac (claset() addSEs [apps_betasE]));
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by (assume_tac 1);
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by (blast_tac (claset() addIs [subst_preserves_beta,apps_preserves_beta]) 1);
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by (blast_tac (claset()
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addIs [apps_preserves_beta2,subst_preserves_beta2,rtrancl_converseI]
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addDs [acc_downwards]) 1);
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(* FIXME: acc_downwards can be replaced by acc(R ^* ) = acc(r) *)
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by (blast_tac (claset() addDs [apps_preserves_betas]) 1);
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qed_spec_mp "double_induction_lemma";
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Goal "t : IT ==> t : termi(beta)";
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by (etac IT.induct 1);
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by (dtac rev_subsetD 1);
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by (rtac lists_mono 1);
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by (rtac Int_lower2 1);
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by (Asm_full_simp_tac 1);
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by (dtac lists_accD 1);
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by (etac acc_induct 1);
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by (rtac accI 1);
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by (blast_tac (claset() addSDs [head_Var_reduction]) 1);
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by (etac acc_induct 1);
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by (rtac accI 1);
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by (Blast_tac 1);
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by (blast_tac (claset() addIs [double_induction_lemma]) 1);
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qed "IT_implies_termi";
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(*** Every terminating term is in IT ***)
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val IT_cases = map (IT.mk_cases
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([Var_apps_eq_Var_apps_conv, Abs_eq_apps_conv, apps_eq_Abs_conv,
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Abs_apps_eq_Abs_apps_conv,
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Var_apps_neq_Abs_apps, Var_apps_neq_Abs_apps RS not_sym,
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hd(tl(get_thms List.thy "foldl.simps")) RS sym ]
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@ dB.simps @ list.simps))
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["Var n $$ ss : IT", "Abs t : IT", "Abs r $ s $$ ts : IT"];
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AddSEs IT_cases;
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(* Turned out to be redundant:
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Goal "t : termi beta ==> \
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\ !r rs. t = r$$rs --> r : termi beta & rs : termi(step1 beta)";
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by (etac acc_induct 1);
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by (force_tac (claset()
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addIs [apps_preserves_beta,apps_preserves_betas,accI],simpset()) 1);
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val lemma = result();
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Goal "r$$rs : termi beta ==> r : termi beta & rs : termi(step1 beta)";
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by (dtac lemma 1);
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by (Blast_tac 1);
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qed "apps_in_termi_betaD";
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Goal "t : termi beta ==> !r. t = Abs r --> r : termi beta";
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by (etac acc_induct 1);
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by (force_tac (claset() addIs [accI],simpset()) 1);
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val lemma = result();
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Goal "Abs r : termi beta ==> r : termi beta";
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by (dtac lemma 1);
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by (Blast_tac 1);
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qed "Abs_in_termi_betaD";
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Goal "t : termi beta ==> !r s. t = r$s --> r : termi beta & s : termi beta";
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by (etac acc_induct 1);
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by (force_tac (claset() addIs [accI],simpset()) 1);
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val lemma = result();
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Goal "r$s : termi beta ==> r : termi beta & s : termi beta";
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by (dtac lemma 1);
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by (Blast_tac 1);
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qed "App_in_termi_betaD";
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*)
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Goal "r : termi beta ==> r : IT";
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by (etac acc_induct 1);
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by (rename_tac "r" 1);
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by (etac thin_rl 1);
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by (etac rev_mp 1);
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by (Simp_tac 1);
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by (res_inst_tac [("t","r")] Apps_dB_induct 1);
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by (rename_tac "n ts" 1);
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by (Clarify_tac 1);
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by (resolve_tac IT.intrs 1);
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by (Clarify_tac 1);
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by (EVERY1[dtac bspec, atac]);
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by (etac mp 1);
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by (Clarify_tac 1);
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by (dtac converseI 1);
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by (EVERY1[dtac ex_step1I, atac]);
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by (Clarify_tac 1);
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by (rename_tac "us" 1);
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by (eres_inst_tac [("x","Var n $$ us")] allE 1);
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by (Force_tac 1);
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by (rename_tac "u ts" 1);
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by (exhaust_tac "ts" 1);
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by (Asm_full_simp_tac 1);
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by (blast_tac (claset() addIs IT.intrs) 1);
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by (rename_tac "s ss" 1);
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by (Asm_full_simp_tac 1);
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by (Clarify_tac 1);
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by (resolve_tac IT.intrs 1);
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by (blast_tac (claset() addIs [apps_preserves_beta]) 1);
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by (etac mp 1);
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by (Clarify_tac 1);
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by (rename_tac "t" 1);
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by (eres_inst_tac [("x","Abs u $ t $$ ss")] allE 1);
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by (Force_tac 1);
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qed "termi_implies_IT";
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