header {* Constant definitions *}theory Consts imports Utils beginfun Umn :: "nat \<Rightarrow> nat \<Rightarrow> lam"where [simp del]: "Umn 0 n = \<integral>(cn 0). V (cn n)"| [simp del]: "Umn (Suc m) n = \<integral>(cn (Suc m)). Umn m n"lemma [simp]: "2 = Suc 1" by autolemma Lam_U: "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 0 = \<integral>x. \<integral>y. \<integral>z. V z" "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 1 = \<integral>x. \<integral>y. \<integral>z. V y" "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 2 = \<integral>x. \<integral>y. \<integral>z. V x" apply (simp_all add: Umn.simps Abs1_eq_iff lam.fresh fresh_at_base flip_def[symmetric] Umn.simps) apply (smt Zero_not_Suc cnd flip_at_base_simps flip_at_simps)+ donelemma a: "n \<le> m \<Longrightarrow> atom (cn n) \<notin> supp (Umn m n)" apply (induct m) apply (auto simp add: lam.supp supp_at_base Umn.simps) by smtlemma b: "supp (Umn m n) \<subseteq> {atom (cn n)}" by (induct m) (auto simp add: lam.supp supp_at_base Umn.simps)lemma supp_U[simp]: "n \<le> m \<Longrightarrow> supp (Umn m n) = {}" using a b by blastlemma U_eqvt: "n \<le> m \<Longrightarrow> p \<bullet> (Umn m n) = Umn m n" by (rule_tac [!] perm_supp_eq) (simp_all add: fresh_star_def fresh_def)definition Var where "Var \<equiv> \<integral>cx. \<integral>cy. (V cy \<cdot> (Umn 2 2) \<cdot> V cx \<cdot> V cy)"definition "App \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. (V cz \<cdot> Umn 2 1 \<cdot> V cx \<cdot> V cy \<cdot> V cz)"definition "Abs \<equiv> \<integral>cx. \<integral>cy. (V cy \<cdot> Umn 2 0 \<cdot> V cx \<cdot> V cy)"lemma Var_App_Abs: "x \<noteq> e \<Longrightarrow> Var = \<integral>x. \<integral>e. (V e \<cdot> Umn 2 2 \<cdot> V x \<cdot> V e)" "e \<noteq> x \<Longrightarrow> e \<noteq> y \<Longrightarrow> x \<noteq> y \<Longrightarrow> App = \<integral>x. \<integral>y. \<integral>e. (V e \<cdot> Umn 2 1 \<cdot> V x \<cdot> V y \<cdot> V e)" "x \<noteq> e \<Longrightarrow> Abs = \<integral>x. \<integral>e. (V e \<cdot> Umn 2 0 \<cdot> V x \<cdot> V e)" unfolding Var_def App_def Abs_def by (simp_all add: Abs1_eq_iff lam.fresh flip_def[symmetric] U_eqvt fresh_def lam.supp supp_at_base) (smt cx_cy_cz permute_flip_at Zero_not_Suc cnd flip_at_base_simps flip_at_simps)+lemma Var_app: "Var \<cdot> x \<cdot> e \<approx> e \<cdot> Umn 2 2 \<cdot> x \<cdot> e" by (rule lam2_fast_app) (simp_all add: Var_App_Abs)lemma App_app: "App \<cdot> x \<cdot> y \<cdot> e \<approx> e \<cdot> Umn 2 1 \<cdot> x \<cdot> y \<cdot> e" by (rule lam3_fast_app[OF Var_App_Abs(2)]) (simp_all)lemma Abs_app: "Abs \<cdot> x \<cdot> e \<approx> e \<cdot> Umn 2 0 \<cdot> x \<cdot> e" by (rule lam2_fast_app) (simp_all add: Var_App_Abs)lemma supp_Var_App_Abs[simp]: "supp Var = {}" "supp App = {}" "supp Abs = {}" by (simp_all add: Var_def App_def Abs_def lam.supp supp_at_base) blast+lemma Var_App_Abs_eqvt[eqvt]: "p \<bullet> Var = Var" "p \<bullet> App = App" "p \<bullet> Abs = Abs" by (rule_tac [!] perm_supp_eq) (simp_all add: fresh_star_def fresh_def)nominal_primrec Numeral :: "lam \<Rightarrow> lam" ("\<lbrace>_\<rbrace>" 1000)where "\<lbrace>V x\<rbrace> = Var \<cdot> (V x)"| Ap: "\<lbrace>M \<cdot> N\<rbrace> = App \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>"| "\<lbrace>\<integral>x. M\<rbrace> = Abs \<cdot> (\<integral>x. \<lbrace>M\<rbrace>)"proof auto fix x :: lam and P assume "\<And>xa. x = V xa \<Longrightarrow> P" "\<And>M N. x = M \<cdot> N \<Longrightarrow> P" "\<And>xa M. x = \<integral> xa. M \<Longrightarrow> P" then show "P" by (rule_tac y="x" and c="0 :: perm" in lam.strong_exhaust) (auto simp add: Abs1_eq_iff fresh_star_def)[3]next fix x :: var and M and xa :: var and Ma assume "[[atom x]]lst. M = [[atom xa]]lst. Ma" "eqvt_at Numeral_sumC M" then show "[[atom x]]lst. Numeral_sumC M = [[atom xa]]lst. Numeral_sumC Ma" apply - apply (erule Abs_lst1_fcb) apply (simp_all add: Abs_fresh_iff) apply (erule fresh_eqvt_at) apply (simp_all add: finite_supp Abs1_eq_iff eqvt_at_def) donenext show "eqvt Numeral_graph" unfolding eqvt_def Numeral_graph_def by (rule, perm_simp, rule)qedtermination (eqvt) by lexicographic_orderlemma supp_numeral[simp]: "supp \<lbrace>x\<rbrace> = supp x" by (induct x rule: lam.induct) (simp_all add: lam.supp)lemma fresh_numeral[simp]: "x \<sharp> \<lbrace>y\<rbrace> = x \<sharp> y" unfolding fresh_def by simpfun app_lst :: "var \<Rightarrow> lam list \<Rightarrow> lam" where "app_lst n [] = V n"| "app_lst n (h # t) = Ap (app_lst n t) h"lemma app_lst_eqvt[eqvt]: "p \<bullet> (app_lst t ts) = app_lst (p \<bullet> t) (p \<bullet> ts)" by (induct ts arbitrary: t p) (simp_all add: eqvts)lemma supp_app_lst: "supp (app_lst x l) = {atom x} \<union> supp l" apply (induct l) apply (simp_all add: supp_Nil lam.supp supp_at_base supp_Cons) by blastlemma app_lst_eq_iff: "app_lst n M = app_lst n N \<Longrightarrow> M = N" by (induct M N rule: list_induct2') simp_alllemma app_lst_rev_eq_iff: "app_lst n (rev M) = app_lst n (rev N) \<Longrightarrow> M = N" by (drule app_lst_eq_iff) simpnominal_primrec Ltgt :: "lam list \<Rightarrow> lam" ("\<guillemotleft>_\<guillemotright>" 1000)where [simp del]: "atom x \<sharp> l \<Longrightarrow> \<guillemotleft>l\<guillemotright> = \<integral>x. (app_lst x (rev l))" unfolding eqvt_def Ltgt_graph_def apply (rule, perm_simp, rule, rule) apply (rule_tac x="x" and ?'a="var" in obtain_fresh) apply (simp_all add: Abs1_eq_iff lam.fresh swap_fresh_fresh fresh_at_base) apply (simp add: eqvts swap_fresh_fresh) apply (case_tac "x = xa") apply simp_all apply (subgoal_tac "eqvt app_lst") apply (erule fresh_fun_eqvt_app2) apply (simp_all add: fresh_at_base lam.fresh eqvt_def eqvts_raw fresh_rev) donetermination (eqvt) by lexicographic_orderlemma ltgt_eq_iff[simp]: "\<guillemotleft>M\<guillemotright> = \<guillemotleft>N\<guillemotright> \<longleftrightarrow> M = N"proof auto obtain x :: var where "atom x \<sharp> (M, N)" using obtain_fresh by auto then have *: "atom x \<sharp> M" "atom x \<sharp> N" using fresh_Pair by simp_all then show "(\<guillemotleft>M\<guillemotright> = \<guillemotleft>N\<guillemotright>) \<Longrightarrow> (M = N)" by (simp add: Abs1_eq_iff app_lst_rev_eq_iff Ltgt.simps)qedlemma Ltgt1_app: "\<guillemotleft>[M]\<guillemotright> \<cdot> N \<approx> N \<cdot> M"proof - obtain x :: var where "atom x \<sharp> (M, N)" using obtain_fresh by auto then have "atom x \<sharp> M" "atom x \<sharp> N" using fresh_Pair by simp_all then show ?thesis apply (subst Ltgt.simps) apply (simp add: fresh_Cons fresh_Nil) apply (rule b3, rule bI, simp add: b1) doneqedlemma Ltgt3_app: "\<guillemotleft>[M,N,P]\<guillemotright> \<cdot> R \<approx> R \<cdot> M \<cdot> N \<cdot> P"proof - obtain x :: var where "atom x \<sharp> (M, N, P, R)" using obtain_fresh by auto then have *: "atom x \<sharp> (M,N,P)" "atom x \<sharp> R" using fresh_Pair by simp_all then have s: "V x \<cdot> M \<cdot> N \<cdot> P [x ::= R] \<approx> R \<cdot> M \<cdot> N \<cdot> P" using b1 by simp show ?thesis using * apply (subst Ltgt.simps) apply (simp add: fresh_Cons fresh_Nil fresh_Pair_elim) apply auto[1] apply (rule b3, rule bI, simp add: b1) doneqedlemma supp_ltgt[simp]: "supp \<guillemotleft>t\<guillemotright> = supp t"proof - obtain x :: var where *:"atom x \<sharp> t" using obtain_fresh by auto show ?thesis using * by (simp_all add: Ltgt.simps lam.supp supp_at_base supp_Nil supp_app_lst supp_rev fresh_def)qedlemma fresh_ltgt[simp]: "x \<sharp> \<guillemotleft>[y]\<guillemotright> = x \<sharp> y" "x \<sharp> \<guillemotleft>[t,r,s]\<guillemotright> = x \<sharp> (t,r,s)" by (simp_all add: fresh_def supp_Cons supp_Nil supp_Pair)lemma Ltgt1_subst[simp]: "\<guillemotleft>[M]\<guillemotright> [y ::= A] = \<guillemotleft>[M [y ::= A]]\<guillemotright>"proof - obtain x :: var where a: "atom x \<sharp> (M, A, y, M [y ::= A])" using obtain_fresh by blast have "x \<noteq> y" using a[simplified fresh_Pair fresh_at_base] by simp then show ?thesis apply (subst Ltgt.simps) using a apply (simp add: fresh_Nil fresh_Cons fresh_Pair_elim) apply (subst Ltgt.simps) using a apply (simp add: fresh_Pair_elim fresh_Nil fresh_Cons) apply (simp add: a) doneqedlemma U_app: "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 2 \<approx> A" "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 1 \<approx> B" "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 0 \<approx> C" by (rule b3, rule Ltgt3_app, rule lam3_fast_app, rule Lam_U, simp_all) (rule b3, rule Ltgt3_app, rule lam3_fast_app, rule Lam_U[simplified], simp_all)+definition "F1 \<equiv> \<integral>cx. (App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> V cx))"definition "F2 \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. ((App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (V cz \<cdot> V cx))) \<cdot> (V cz \<cdot> V cy))"definition "F3 \<equiv> \<integral>cx. \<integral>cy. (App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>cz. (V cy \<cdot> (V cx \<cdot> V cz)))))"lemma Lam_F: "F1 = \<integral>x. (App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> V x))" "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> c \<noteq> b \<Longrightarrow> F2 = \<integral>a. \<integral>b. \<integral>c. ((App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (V c \<cdot> V a))) \<cdot> (V c \<cdot> V b))" "a \<noteq> b \<Longrightarrow> a \<noteq> x \<Longrightarrow> x \<noteq> b \<Longrightarrow> F3 = \<integral>a. \<integral>b. (App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>x. (V b \<cdot> (V a \<cdot> V x)))))" apply (simp_all add: F1_def F2_def F3_def Abs1_eq_iff lam.fresh supp_at_base Var_App_Abs_eqvt Numeral.eqvt flip_def[symmetric] fresh_at_base) apply (smt cx_cy_cz permute_flip_at)+ donelemma supp_F[simp]: "supp F1 = {}" "supp F2 = {}" "supp F3 = {}" by (simp_all add: F1_def F2_def F3_def lam.supp supp_at_base) blast+lemma F_eqvt[eqvt]: "p \<bullet> F1 = F1" "p \<bullet> F2 = F2" "p \<bullet> F3 = F3" by (rule_tac [!] perm_supp_eq) (simp_all add: fresh_star_def fresh_def)lemma F_app: "F1 \<cdot> A \<approx> App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> A)" "F2 \<cdot> A \<cdot> B \<cdot> C \<approx> (App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (C \<cdot> A))) \<cdot> (C \<cdot> B)" by (rule lam1_fast_app, rule Lam_F, simp_all) (rule lam3_fast_app, rule Lam_F, simp_all)lemma F3_app: assumes f: "atom x \<sharp> A" "atom x \<sharp> B" (* or A and B have empty support *) shows "F3 \<cdot> A \<cdot> B \<approx> App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>x. (B \<cdot> (A \<cdot> V x))))"proof - obtain y :: var where b: "atom y \<sharp> (x, A, B)" using obtain_fresh by blast obtain z :: var where c: "atom z \<sharp> (x, y, A, B)" using obtain_fresh by blast have *: "x \<noteq> z" "x \<noteq> y" "y \<noteq> z" using b c by (simp_all add: fresh_Pair fresh_at_base) blast+ have **: "atom y \<sharp> z" "atom x \<sharp> z" "atom y \<sharp> x" "atom z \<sharp> y" "atom z \<sharp> x" "atom x \<sharp> y" "atom x \<sharp> A" "atom y \<sharp> A" "atom z \<sharp> A" "atom x \<sharp> B" "atom y \<sharp> B" "atom z \<sharp> B" using b c f by (simp_all add: fresh_Pair fresh_at_base) blast+ show ?thesis apply (simp add: Lam_F(3)[of y z x] * *[symmetric]) apply (rule b3) apply (rule b5) apply (rule bI) apply (simp add: ** fresh_Pair * *[symmetric]) apply (rule b3) apply (rule bI) apply (simp add: ** fresh_Pair * *[symmetric]) apply (rule b1) doneqeddefinition Lam_A1_pre : "A1 \<equiv> \<integral>cx. \<integral>cy. (F1 \<cdot> V cx)"definition Lam_A2_pre : "A2 \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. (F2 \<cdot> V cx \<cdot> V cy \<cdot> \<guillemotleft>[V cz]\<guillemotright>)"definition Lam_A3_pre : "A3 \<equiv> \<integral>cx. \<integral>cy. (F3 \<cdot> V cx \<cdot> \<guillemotleft>[V cy]\<guillemotright>)"lemma Lam_A: "x \<noteq> y \<Longrightarrow> A1 = \<integral>x. \<integral>y. (F1 \<cdot> V x)" "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> c \<noteq> b \<Longrightarrow> A2 = \<integral>a. \<integral>b. \<integral>c. (F2 \<cdot> V a \<cdot> V b \<cdot> \<guillemotleft>[V c]\<guillemotright>)" "a \<noteq> b \<Longrightarrow> A3 = \<integral>a. \<integral>b. (F3 \<cdot> V a \<cdot> \<guillemotleft>[V b]\<guillemotright>)" apply (simp_all add: Lam_A1_pre Lam_A2_pre Lam_A3_pre Abs1_eq_iff lam.fresh supp_at_base Var_App_Abs_eqvt Numeral.eqvt flip_def[symmetric] fresh_at_base F_eqvt Ltgt.eqvt) apply (smt cx_cy_cz permute_flip_at)+ donelemma supp_A[simp]: "supp A1 = {}" "supp A2 = {}" "supp A3 = {}" by (auto simp add: Lam_A1_pre Lam_A2_pre Lam_A3_pre lam.supp supp_at_base supp_Cons supp_Nil)lemma A_app: "A1 \<cdot> A \<cdot> B \<approx> F1 \<cdot> A" "A2 \<cdot> A \<cdot> B \<cdot> C \<approx> F2 \<cdot> A \<cdot> B \<cdot> \<guillemotleft>[C]\<guillemotright>" "A3 \<cdot> A \<cdot> B \<approx> F3 \<cdot> A \<cdot> \<guillemotleft>[B]\<guillemotright>" apply (rule lam2_fast_app, rule Lam_A, simp_all) apply (rule lam3_fast_app, rule Lam_A, simp_all) apply (rule lam2_fast_app, rule Lam_A, simp_all) donedefinition "Num \<equiv> \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright>"lemma supp_Num[simp]: "supp Num = {}" by (auto simp only: Num_def supp_ltgt supp_Pair supp_A supp_Cons supp_Nil)end