# HG changeset patch # User Christian Urban # Date 1309304930 -3600 # Node ID a27215ab674e247a115c82a57905fff3811d712f # Parent 6b496f69f76c516477299a95bc38f2e33a81b537 some experiments diff -r 6b496f69f76c -r a27215ab674e Nominal/Ex/Let.thy --- a/Nominal/Ex/Let.thy Tue Jun 28 14:45:30 2011 +0900 +++ b/Nominal/Ex/Let.thy Wed Jun 29 00:48:50 2011 +0100 @@ -18,9 +18,12 @@ "bn ANil = []" | "bn (ACons x t as) = (atom x) # (bn as)" +print_theorems + thm trm_assn.fv_defs thm trm_assn.eq_iff thm trm_assn.bn_defs +thm trm_assn.bn_inducts thm trm_assn.perm_simps thm trm_assn.induct thm trm_assn.inducts @@ -30,6 +33,17 @@ thm trm_assn.exhaust thm trm_assn.strong_exhaust +lemma bn_inj: + assumes a: "alpha_bn_raw x y" + shows "bn_raw x = bn_raw y \ x = y" +using a +apply(induct) +apply(auto)[6] +apply(simp) +apply(simp) +oops + + lemma lets_bla: "x \ z \ y \ z \ x \ y \(Let (ACons x (Var y) ANil) (Var x)) \ (Let (ACons x (Var z) ANil) (Var x))" @@ -96,17 +110,19 @@ (* TODO: should be provided by nominal *) lemmas [eqvt] = trm_assn.fv_bn_eqvt +thm Abs_lst_fcb + +(* lemma Abs_lst_fcb2: fixes as bs :: "'a :: fs" and x y :: "'b :: fs" and c::"'c::fs" assumes eq: "[ba as]lst. x = [ba bs]lst. y" - and fcb1: "(set (ba as)) \* f as x c" + and fcb1: "set (ba as) \* f as x c" and fresh1: "set (ba as) \* c" and fresh2: "set (ba bs) \* c" and perm1: "\p. supp p \* c \ p \ (f as x c) = f (p \ as) (p \ x) c" and perm2: "\p. supp p \* c \ p \ (f bs y c) = f (p \ bs) (p \ y) c" - and props: "eqvt ba" "inj ba" shows "f as x c = f bs y c" proof - have "supp (as, x, c) supports (f as x c)" @@ -123,8 +139,8 @@ fr1: "(q \ (set (ba as))) \* (x, c, f as x c, f bs y c)" and fr2: "supp q \* ([ba as]lst. x)" and inc: "supp q \ (set (ba as)) \ q \ (set (ba as))" - using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" and x="[ba as]lst. x"] - fin1 fin2 + using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" + and x="[ba as]lst. x"] fin1 fin2 by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) have "[q \ (ba as)]lst. (q \ x) = q \ ([ba as]lst. x)" by simp also have "\ = [ba as]lst. x" @@ -142,10 +158,6 @@ apply(simp add: set_eqvt) apply(blast) done - have qq4: "q \ as = r \ bs" using qq2 props unfolding eqvt_def inj_on_def - apply(perm_simp) - apply(simp) - done have "(set (ba as)) \* f as x c" by (rule fcb1) then have "q \ ((set (ba as)) \* f as x c)" by (simp add: permute_bool_def) @@ -155,32 +167,32 @@ using inc fresh1 fr1 apply(auto simp add: fresh_star_def fresh_Pair) done - then have "set (r \ (ba bs)) \* f (r \ bs) (r \ y) c" using qq1 qq2 qq4 + then have "set (r \ (ba bs)) \* f (r \ bs) (r \ y) c" using qq1 qq2 by simp - then have "r \ ((set (ba bs)) \* f bs y c)" + then have "r \ ((set (ba bs)) \* f (ba bs) y c)" apply(simp add: fresh_star_eqvt set_eqvt) apply(subst (asm) perm2[symmetric]) using qq3 fresh2 fr1 apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) done - then have fcb2: "(set (ba bs)) \* f bs y c" by (simp add: permute_bool_def) - have "f as x c = q \ (f as x c)" + then have fcb2: "(set (ba bs)) \* f (ba bs) y c" by (simp add: permute_bool_def) + have "f (ba as) x c = q \ (f (ba as) x c)" apply(rule perm_supp_eq[symmetric]) using inc fcb1 fr1 by (auto simp add: fresh_star_def) - also have "\ = f (q \ as) (q \ x) c" + also have "\ = f (q \ (ba as)) (q \ x) c" apply(rule perm1) using inc fresh1 fr1 by (auto simp add: fresh_star_def) - also have "\ = f (r \ bs) (r \ y) c" using qq1 qq4 by simp - also have "\ = r \ (f bs y c)" + also have "\ = f (r \ (ba bs)) (r \ y) c" using qq1 qq2 by simp + also have "\ = r \ (f (ba bs) y c)" apply(rule perm2[symmetric]) using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) - also have "... = f bs y c" + also have "... = f (ba bs) y c" apply(rule perm_supp_eq) using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) finally show ?thesis by simp qed +*) -(* PROBLEM: the proof needs induction on alpha_bn inside which is not possible... *) nominal_primrec height_trm :: "trm \ nat" and height_assn :: "assn \ nat" @@ -200,13 +212,42 @@ apply (drule_tac x="trm" in meta_spec) apply (simp add: alpha_bn_refl) apply (case_tac b rule: trm_assn.exhaust(2)) - apply (auto) - apply (erule Abs_lst1_fcb) - apply (simp_all add: pure_fresh) + apply (auto)[2] + apply(simp_all) + thm trm_assn.perm_bn_alpha trm_assn.permute_bn + apply (erule_tac c="()" in Abs_lst_fcb2) + apply (simp_all add: pure_fresh fresh_star_def)[3] + apply (simp add: eqvt_at_def) apply (simp add: eqvt_at_def) - apply (erule Abs_lst_fcb) - apply (simp_all add: pure_fresh) - apply (simp_all add: eqvt_at_def eqvts) + apply(erule conjE) + apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff]) + apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff]) + apply (subgoal_tac "eqvt_at height_assn as") + apply (subgoal_tac "eqvt_at height_assn asa") + apply (subgoal_tac "eqvt_at height_trm b") + apply (subgoal_tac "eqvt_at height_trm ba") + apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)") + apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)") + apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)") + apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)") + defer + apply (simp add: eqvt_at_def height_trm_def) + apply (simp add: eqvt_at_def height_trm_def) + apply (simp add: eqvt_at_def height_assn_def) + apply (simp add: eqvt_at_def height_assn_def) + apply (subgoal_tac "height_assn as = height_assn asa") + apply (subgoal_tac "height_trm b = height_trm ba") + apply simp + apply (erule_tac c="()" in Abs_lst_fcb2) + apply (simp_all add: pure_fresh fresh_star_def)[3] + apply (simp_all add: eqvt_at_def)[2] + apply (drule_tac c="()" in Abs_lst_fcb2) + apply (simp_all add: pure_fresh fresh_star_def)[3] + apply (simp_all add: eqvt_at_def)[2] + apply(simp add: eqvt_def) + apply(perm_simp) + apply(simp) + apply(simp add: inj_on_def) apply (rule arg_cong) back oops diff -r 6b496f69f76c -r a27215ab674e Nominal/Ex/LetRecB.thy --- a/Nominal/Ex/LetRecB.thy Tue Jun 28 14:45:30 2011 +0900 +++ b/Nominal/Ex/LetRecB.thy Wed Jun 29 00:48:50 2011 +0100 @@ -181,6 +181,11 @@ apply (simp add: eqvt_at_def) apply (simp add: eqvt_at_def) --"" + apply(simp_all add: eqvt_def inj_on_def) + apply(perm_simp) + apply(simp) + apply(perm_simp) + apply(simp) done termination by lexicographic_order diff -r 6b496f69f76c -r a27215ab674e Nominal/Nominal2.thy --- a/Nominal/Nominal2.thy Tue Jun 28 14:45:30 2011 +0900 +++ b/Nominal/Nominal2.thy Wed Jun 29 00:48:50 2011 +0100 @@ -218,7 +218,7 @@ val (_, lthy3) = Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_perm_simps) lthy2a val _ = trace_msg (K "Defining raw fv- and bn-functions...") - val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_induct, lthy3a) = + val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_inducts, lthy3a) = define_raw_bns raw_dt_names raw_dts raw_bn_funs raw_bn_eqs (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3 @@ -235,19 +235,22 @@ val (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy4) = define_raw_alpha raw_dt_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3c + val _ = tracing ("alpha_induct\n" ^ Syntax.string_of_term lthy3c (prop_of alpha_induct)) + val _ = tracing ("alpha_intros\n" ^ cat_lines (map (Syntax.string_of_term lthy4 o prop_of) alpha_intros)) + val alpha_tys = map (domain_type o fastype_of) alpha_trms val _ = trace_msg (K "Proving distinct theorems...") val alpha_distincts = - mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_dt_names + raw_prove_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_dt_names val _ = trace_msg (K "Proving eq-iff theorems...") val alpha_eq_iff = - mk_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases + raw_prove_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases val _ = trace_msg (K "Proving equivariance of bns, fvs, size and alpha...") val raw_bn_eqvt = - raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) lthy4 + raw_prove_eqvt raw_bns raw_bn_inducts (raw_bn_defs @ raw_perm_simps) lthy4 (* noting the raw_bn_eqvt lemmas in a temprorary theory *) val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_bn_eqvt) lthy4) @@ -288,6 +291,8 @@ val alpha_bn_imp_thms = raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy5 + val _ = tracing ("alpha_bn_imp_thms:\n" ^ cat_lines (map (Syntax.string_of_term lthy5 o prop_of) alpha_bn_imp_thms)) + val _ = trace_msg (K "Proving respectfulness...") val raw_funs_rsp_aux = raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms raw_fvs @@ -381,7 +386,8 @@ val eq_iff_simps = @{thms alphas permute_prod.simps prod_fv.simps prod_alpha_def prod_rel_def prod.cases} - val ((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), lthyA) = + val (((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), qbn_inducts), + lthyA) = lthy9a |> lift_thms qtys [] alpha_distincts ||>> lift_thms qtys eq_iff_simps alpha_eq_iff @@ -389,6 +395,7 @@ ||>> lift_thms qtys [] raw_bn_defs ||>> lift_thms qtys [] raw_perm_simps ||>> lift_thms qtys [] (raw_fv_eqvt @ raw_bn_eqvt) + ||>> lift_thms qtys [] raw_bn_inducts val ((((((qsize_eqvt, [qinduct]), qexhausts), qsize_simps), qperm_bn_simps), qalpha_refl_thms), lthyB) = lthyA @@ -399,7 +406,7 @@ ||>> lift_thms qtys [] raw_perm_bn_simps ||>> lift_thms qtys [] alpha_refl_thms - val qinducts = Project_Rule.projections lthyA qinduct + val qinducts = Project_Rule.projections lthyB qinduct val _ = trace_msg (K "Proving supp lemmas and fs-instances...") val qsupports_thms = @@ -470,8 +477,9 @@ ||>> Local_Theory.note ((thms_suffix "eq_iff", [induct_attr, simp_attr]), qeq_iffs') ||>> Local_Theory.note ((thms_suffix "fv_defs", []), qfv_defs) ||>> Local_Theory.note ((thms_suffix "bn_defs", []), qbn_defs) + ||>> Local_Theory.note ((thms_suffix "bn_inducts", []), qbn_inducts) ||>> Local_Theory.note ((thms_suffix "perm_simps", [eqvt_attr, simp_attr]), qperm_simps) - ||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", []), qfv_qbn_eqvts) + ||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", [eqvt_attr]), qfv_qbn_eqvts) ||>> Local_Theory.note ((thms_suffix "size", [simp_attr]), qsize_simps) ||>> Local_Theory.note ((thms_suffix "size_eqvt", []), qsize_eqvt) ||>> Local_Theory.note ((thms_suffix "induct", [case_names_attr]), [qinduct]) diff -r 6b496f69f76c -r a27215ab674e Nominal/nominal_dt_alpha.ML --- a/Nominal/nominal_dt_alpha.ML Tue Jun 28 14:45:30 2011 +0900 +++ b/Nominal/nominal_dt_alpha.ML Wed Jun 29 00:48:50 2011 +0100 @@ -13,10 +13,10 @@ bclause list list list -> term list -> Proof.context -> term list * term list * thm list * thm list * thm * local_theory - val mk_alpha_distincts: Proof.context -> thm list -> thm list -> + val raw_prove_alpha_distincts: Proof.context -> thm list -> thm list -> term list -> string list -> thm list - val mk_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> + val raw_prove_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> thm list -> thm list val induct_prove: typ list -> (typ * (term -> term)) list -> thm -> @@ -297,7 +297,7 @@ THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps distinct_thms) -fun mk_alpha_distincts ctxt cases_thms distinct_thms alpha_trms alpha_str = +fun raw_prove_alpha_distincts ctxt cases_thms distinct_thms alpha_trms alpha_str = let val ty_trm_assoc = alpha_str ~~ (map (fst o dest_Const) alpha_trms) @@ -341,7 +341,7 @@ else HOLogic.mk_Trueprop (HOLogic.mk_eq (concl, list_conj hyps)) end; -fun mk_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims = +fun raw_prove_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims = let val ((_, thms_imp), ctxt') = Variable.import false alpha_intros ctxt; val goals = map mk_alpha_eq_iff_goal thms_imp;