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