--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/CoreHaskell.thy Sun May 09 12:38:59 2010 +0100
@@ -0,0 +1,673 @@
+theory CoreHaskell
+imports "../NewParser"
+begin
+
+(* core haskell *)
+
+atom_decl var
+atom_decl cvar
+atom_decl tvar
+
+nominal_datatype tkind =
+ KStar
+| KFun "tkind" "tkind"
+and ckind =
+ CKEq "ty" "ty"
+and ty =
+ TVar "tvar"
+| TC "string"
+| TApp "ty" "ty"
+| TFun "string" "ty_lst"
+| TAll tv::"tvar" "tkind" T::"ty" bind tv in T
+| TEq "ckind" "ty"
+and ty_lst =
+ TsNil
+| TsCons "ty" "ty_lst"
+and co =
+ CVar "cvar"
+| CConst "string"
+| CApp "co" "co"
+| CFun "string" "co_lst"
+| CAll cv::"cvar" "ckind" C::"co" bind cv in C
+| CEq "ckind" "co"
+| CRefl "ty"
+| CSym "co"
+| CCir "co" "co"
+| CAt "co" "ty"
+| CLeft "co"
+| CRight "co"
+| CSim "co" "co"
+| CRightc "co"
+| CLeftc "co"
+| CCoe "co" "co"
+and co_lst =
+ CsNil
+| CsCons "co" "co_lst"
+and trm =
+ Var "var"
+| K "string"
+| LAMT tv::"tvar" "tkind" t::"trm" bind tv in t
+| LAMC cv::"cvar" "ckind" t::"trm" bind cv in t
+| AppT "trm" "ty"
+| AppC "trm" "co"
+| Lam v::"var" "ty" t::"trm" bind v in t
+| App "trm" "trm"
+| Let x::"var" "ty" "trm" t::"trm" bind x in t
+| Case "trm" "assoc_lst"
+| Cast "trm" "ty" --"ty is supposed to be a coercion type only"
+and assoc_lst =
+ ANil
+| ACons p::"pat" t::"trm" "assoc_lst" bind "bv p" in t
+and pat =
+ Kpat "string" "tvars" "cvars" "vars"
+and vars =
+ VsNil
+| VsCons "var" "ty" "vars"
+and tvars =
+ TvsNil
+| TvsCons "tvar" "tkind" "tvars"
+and cvars =
+ CvsNil
+| CvsCons "cvar" "ckind" "cvars"
+binder
+ bv :: "pat \<Rightarrow> atom list"
+and bv_vs :: "vars \<Rightarrow> atom list"
+and bv_tvs :: "tvars \<Rightarrow> atom list"
+and bv_cvs :: "cvars \<Rightarrow> atom list"
+where
+ "bv (Kpat s tvts tvcs vs) = append (bv_tvs tvts) (append (bv_cvs tvcs) (bv_vs vs))"
+| "bv_vs VsNil = []"
+| "bv_vs (VsCons v k t) = (atom v) # bv_vs t"
+| "bv_tvs TvsNil = []"
+| "bv_tvs (TvsCons v k t) = (atom v) # bv_tvs t"
+| "bv_cvs CvsNil = []"
+| "bv_cvs (CvsCons v k t) = (atom v) # bv_cvs t"
+
+lemmas fv_supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.supp(1-9,11,13,15)
+lemmas supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.fv[simplified fv_supp]
+lemmas perm=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.perm
+lemmas eq_iff=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.eq_iff
+lemmas inducts=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.inducts
+
+lemmas alpha_inducts=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vars_raw_alpha_tvars_raw_alpha_cvars_raw_alpha_bv_raw_alpha_bv_vs_raw_alpha_bv_tvs_raw_alpha_bv_cvs_raw.inducts
+lemmas alpha_intros=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vars_raw_alpha_tvars_raw_alpha_cvars_raw_alpha_bv_raw_alpha_bv_vs_raw_alpha_bv_tvs_raw_alpha_bv_cvs_raw.intros
+
+lemma fresh_star_minus_perm: "as \<sharp>* - p = as \<sharp>* (p :: perm)"
+ unfolding fresh_star_def Ball_def
+ by auto (simp_all add: fresh_minus_perm)
+
+primrec permute_bv_vs_raw
+where "permute_bv_vs_raw p VsNil_raw = VsNil_raw"
+| "permute_bv_vs_raw p (VsCons_raw v t l) = VsCons_raw (p \<bullet> v) t (permute_bv_vs_raw p l)"
+primrec permute_bv_cvs_raw
+where "permute_bv_cvs_raw p CvsNil_raw = CvsNil_raw"
+| "permute_bv_cvs_raw p (CvsCons_raw v t l) = CvsCons_raw (p \<bullet> v) t (permute_bv_cvs_raw p l)"
+primrec permute_bv_tvs_raw
+where "permute_bv_tvs_raw p TvsNil_raw = TvsNil_raw"
+| "permute_bv_tvs_raw p (TvsCons_raw v t l) = TvsCons_raw (p \<bullet> v) t (permute_bv_tvs_raw p l)"
+primrec permute_bv_raw
+where "permute_bv_raw p (Kpat_raw c l1 l2 l3) =
+ Kpat_raw c (permute_bv_tvs_raw p l1) (permute_bv_cvs_raw p l2) (permute_bv_vs_raw p l3)"
+
+quotient_definition "permute_bv_vs :: perm \<Rightarrow> vars \<Rightarrow> vars"
+is "permute_bv_vs_raw"
+quotient_definition "permute_bv_cvs :: perm \<Rightarrow> cvars \<Rightarrow> cvars"
+is "permute_bv_cvs_raw"
+quotient_definition "permute_bv_tvs :: perm \<Rightarrow> tvars \<Rightarrow> tvars"
+is "permute_bv_tvs_raw"
+quotient_definition "permute_bv :: perm \<Rightarrow> pat \<Rightarrow> pat"
+is "permute_bv_raw"
+
+lemma rsp_pre:
+ "alpha_tvars_raw d a \<Longrightarrow> alpha_tvars_raw (permute_bv_tvs_raw x d) (permute_bv_tvs_raw x a)"
+ "alpha_cvars_raw e b \<Longrightarrow> alpha_cvars_raw (permute_bv_cvs_raw x e) (permute_bv_cvs_raw x b)"
+ "alpha_vars_raw f c \<Longrightarrow> alpha_vars_raw (permute_bv_vs_raw x f) (permute_bv_vs_raw x c)"
+ apply (erule_tac [!] alpha_inducts)
+ apply (simp_all only: alpha_intros perm permute_bv_tvs_raw.simps permute_bv_cvs_raw.simps permute_bv_vs_raw.simps)
+ done
+
+lemma [quot_respect]:
+ "(op = ===> alpha_pat_raw ===> alpha_pat_raw) permute_bv_raw permute_bv_raw"
+ "(op = ===> alpha_tvars_raw ===> alpha_tvars_raw) permute_bv_tvs_raw permute_bv_tvs_raw"
+ "(op = ===> alpha_cvars_raw ===> alpha_cvars_raw) permute_bv_cvs_raw permute_bv_cvs_raw"
+ "(op = ===> alpha_vars_raw ===> alpha_vars_raw) permute_bv_vs_raw permute_bv_vs_raw"
+ apply (simp_all add: rsp_pre)
+ apply clarify
+ apply (erule_tac alpha_inducts)
+ apply (simp_all)
+ apply (rule alpha_intros)
+ apply (simp_all add: rsp_pre)
+ done
+
+thm permute_bv_raw.simps[no_vars]
+ permute_bv_vs_raw.simps[quot_lifted]
+ permute_bv_cvs_raw.simps[quot_lifted]
+ permute_bv_tvs_raw.simps[quot_lifted]
+
+lemma permute_bv_pre:
+ "permute_bv p (Kpat c l1 l2 l3) =
+ Kpat c (permute_bv_tvs p l1) (permute_bv_cvs p l2) (permute_bv_vs p l3)"
+ by (lifting permute_bv_raw.simps)
+
+lemmas permute_bv[simp] =
+ permute_bv_pre
+ permute_bv_vs_raw.simps[quot_lifted]
+ permute_bv_cvs_raw.simps[quot_lifted]
+ permute_bv_tvs_raw.simps[quot_lifted]
+
+lemma perm_bv1:
+ "p \<bullet> bv_cvs b = bv_cvs (permute_bv_cvs p b)"
+ "p \<bullet> bv_tvs c = bv_tvs (permute_bv_tvs p c)"
+ "p \<bullet> bv_vs d = bv_vs (permute_bv_vs p d)"
+ apply(induct b rule: inducts(12))
+ apply(simp_all add:permute_bv eqvts)
+ apply(induct c rule: inducts(11))
+ apply(simp_all add:permute_bv eqvts)
+ apply(induct d rule: inducts(10))
+ apply(simp_all add:permute_bv eqvts)
+ done
+
+lemma perm_bv2:
+ "p \<bullet> bv l = bv (permute_bv p l)"
+ apply(induct l rule: inducts(9))
+ apply(simp_all add:permute_bv)
+ apply(simp add: perm_bv1[symmetric])
+ apply(simp add: eqvts)
+ done
+
+lemma alpha_perm_bn1:
+ "alpha_bv_tvs tvars (permute_bv_tvs q tvars)"
+ "alpha_bv_cvs cvars (permute_bv_cvs q cvars)"
+ "alpha_bv_vs vars (permute_bv_vs q vars)"
+ apply(induct tvars rule: inducts(11))
+ apply(simp_all add:permute_bv eqvts eq_iff)
+ apply(induct cvars rule: inducts(12))
+ apply(simp_all add:permute_bv eqvts eq_iff)
+ apply(induct vars rule: inducts(10))
+ apply(simp_all add:permute_bv eqvts eq_iff)
+ done
+
+lemma alpha_perm_bn:
+ "alpha_bv pat (permute_bv q pat)"
+ apply(induct pat rule: inducts(9))
+ apply(simp_all add:permute_bv eqvts eq_iff alpha_perm_bn1)
+ done
+
+lemma ACons_subst:
+ "supp (Abs_lst (bv pat) trm) \<sharp>* q \<Longrightarrow> (ACons pat trm al) = ACons (permute_bv q pat) (q \<bullet> trm) al"
+ apply (simp only: eq_iff)
+ apply (simp add: alpha_perm_bn)
+ apply (rule_tac x="q" in exI)
+ apply (simp add: alphas)
+ apply (simp add: perm_bv2[symmetric])
+ apply (simp add: eqvts[symmetric])
+ apply (simp add: supp_abs)
+ apply (simp add: fv_supp)
+ apply (rule supp_perm_eq[symmetric])
+ apply (subst supp_finite_atom_set)
+ apply (rule finite_Diff)
+ apply (simp add: finite_supp)
+ apply (assumption)
+ done
+
+lemma permute_bv_zero1:
+ "permute_bv_cvs 0 b = b"
+ "permute_bv_tvs 0 c = c"
+ "permute_bv_vs 0 d = d"
+ apply(induct b rule: inducts(12))
+ apply(simp_all add:permute_bv eqvts)
+ apply(induct c rule: inducts(11))
+ apply(simp_all add:permute_bv eqvts)
+ apply(induct d rule: inducts(10))
+ apply(simp_all add:permute_bv eqvts)
+ done
+
+lemma permute_bv_zero2:
+ "permute_bv 0 a = a"
+ apply(induct a rule: inducts(9))
+ apply(simp_all add:permute_bv eqvts permute_bv_zero1)
+ done
+
+lemma fv_alpha1: "fv_bv_tvs x \<sharp>* pa \<Longrightarrow> alpha_bv_tvs (pa \<bullet> x) x"
+ apply (induct x rule: inducts(11))
+ apply (simp_all add: eq_iff fresh_star_union)
+ done
+
+lemma fv_alpha2: "fv_bv_cvs x \<sharp>* pa \<Longrightarrow> alpha_bv_cvs (pa \<bullet> x) x"
+apply (induct x rule: inducts(12))
+apply (rule TrueI)+
+apply (simp_all add: eq_iff fresh_star_union)
+apply (subst supp_perm_eq)
+apply (simp_all add: fv_supp)
+done
+
+lemma fv_alpha3: "fv_bv_vs x \<sharp>* pa \<Longrightarrow> alpha_bv_vs (pa \<bullet> x) x"
+apply (induct x rule: inducts(10))
+apply (rule TrueI)+
+apply (simp_all add: fresh_star_union eq_iff)
+apply (subst supp_perm_eq)
+apply (simp_all add: fv_supp)
+done
+
+lemma fv_alpha: "fv_bv x \<sharp>* pa \<Longrightarrow> alpha_bv (pa \<bullet> x) x"
+apply (induct x rule: inducts(9))
+apply (rule TrueI)+
+apply (simp_all add: eq_iff fresh_star_union)
+apply (simp add: fv_alpha1 fv_alpha2 fv_alpha3)
+apply (subst supp_perm_eq)
+apply (simp_all add: fv_supp)
+done
+
+lemma fin1: "finite (fv_bv_tvs x)"
+apply (induct x rule: inducts(11))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma fin2: "finite (fv_bv_cvs x)"
+apply (induct x rule: inducts(12))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma fin3: "finite (fv_bv_vs x)"
+apply (induct x rule: inducts(10))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma fin_fv_bv: "finite (fv_bv x)"
+apply (induct x rule: inducts(9))
+apply (rule TrueI)+
+defer
+apply (rule TrueI)+
+apply (simp add: fin1 fin2 fin3)
+apply (rule finite_supp)
+done
+
+lemma finb1: "finite (set (bv_tvs x))"
+apply (induct x rule: inducts(11))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma finb2: "finite (set (bv_cvs x))"
+apply (induct x rule: inducts(12))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma finb3: "finite (set (bv_vs x))"
+apply (induct x rule: inducts(10))
+apply (simp_all add: fv_supp finite_supp)
+done
+
+lemma fin_bv: "finite (set (bv x))"
+apply (induct x rule: inducts(9))
+apply (simp_all add: finb1 finb2 finb3)
+done
+
+lemma strong_induction_principle:
+ assumes a01: "\<And>b. P1 b KStar"
+ and a02: "\<And>tkind1 tkind2 b. \<lbrakk>\<And>c. P1 c tkind1; \<And>c. P1 c tkind2\<rbrakk> \<Longrightarrow> P1 b (KFun tkind1 tkind2)"
+ and a03: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P2 b (CKEq ty1 ty2)"
+ and a04: "\<And>tvar b. P3 b (TVar tvar)"
+ and a05: "\<And>string b. P3 b (TC string)"
+ and a06: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P3 b (TApp ty1 ty2)"
+ and a07: "\<And>string ty_lst b. \<lbrakk>\<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P3 b (TFun string ty_lst)"
+ and a08: "\<And>tvar tkind ty b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P3 c ty; atom tvar \<sharp> b\<rbrakk>
+ \<Longrightarrow> P3 b (TAll tvar tkind ty)"
+ and a09: "\<And>ck ty b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P3 b (TEq ck ty)"
+ and a10: "\<And>b. P4 b TsNil"
+ and a11: "\<And>ty ty_lst b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P4 b (TsCons ty ty_lst)"
+ and a12: "\<And>string b. P5 b (CVar string)"
+ and a12a:"\<And>str b. P5 b (CConst str)"
+ and a13: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CApp co1 co2)"
+ and a14: "\<And>string co_lst b. \<lbrakk>\<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P5 b (CFun string co_lst)"
+ and a15: "\<And>tvar ckind co b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P5 c co; atom tvar \<sharp> b\<rbrakk>
+ \<Longrightarrow> P5 b (CAll tvar ckind co)"
+ and a16: "\<And>ck co b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CEq ck co)"
+ and a17: "\<And>ty b. \<lbrakk>\<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CRefl ty)"
+ and a17a: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CSym co)"
+ and a18: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCir co1 co2)"
+ and a18a:"\<And>co ty b. \<lbrakk>\<And>c. P5 c co; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CAt co ty)"
+ and a19: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeft co)"
+ and a20: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRight co)"
+ and a21: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CSim co1 co2)"
+ and a22: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRightc co)"
+ and a23: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeftc co)"
+ and a24: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCoe co1 co2)"
+ and a25: "\<And>b. P6 b CsNil"
+ and a26: "\<And>co co_lst b. \<lbrakk>\<And>c. P5 c co; \<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P6 b (CsCons co co_lst)"
+ and a27: "\<And>var b. P7 b (Var var)"
+ and a28: "\<And>string b. P7 b (K string)"
+ and a29: "\<And>tvar tkind trm b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>
+ \<Longrightarrow> P7 b (LAMT tvar tkind trm)"
+ and a30: "\<And>tvar ckind trm b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>
+ \<Longrightarrow> P7 b (LAMC tvar ckind trm)"
+ and a31: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (AppT trm ty)"
+ and a31a:"\<And>trm co b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P7 b (AppC trm co)"
+ and a32: "\<And>var ty trm b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm; atom var \<sharp> b\<rbrakk> \<Longrightarrow> P7 b (Lam var ty trm)"
+ and a33: "\<And>trm1 trm2 b. \<lbrakk>\<And>c. P7 c trm1; \<And>c. P7 c trm2\<rbrakk> \<Longrightarrow> P7 b (App trm1 trm2)"
+ and a34: "\<And>var ty trm1 trm2 b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm1; \<And>c. P7 c trm2; atom var \<sharp> b\<rbrakk>
+ \<Longrightarrow> P7 b (Let var ty trm1 trm2)"
+ and a35: "\<And>trm assoc_lst b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P8 c assoc_lst\<rbrakk> \<Longrightarrow> P7 b (Case trm assoc_lst)"
+ and a36: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (Cast trm ty)"
+ and a37: "\<And>b. P8 b ANil"
+ and a38: "\<And>pat trm assoc_lst b. \<lbrakk>\<And>c. P9 c pat; \<And>c. P7 c trm; \<And>c. P8 c assoc_lst; set (bv (pat)) \<sharp>* b\<rbrakk>
+ \<Longrightarrow> P8 b (ACons pat trm assoc_lst)"
+ and a39: "\<And>string tvars cvars vars b. \<lbrakk>\<And>c. P11 c tvars; \<And>c. P12 c cvars; \<And>c. P10 c vars\<rbrakk>
+ \<Longrightarrow> P9 b (Kpat string tvars cvars vars)"
+ and a40: "\<And>b. P10 b VsNil"
+ and a41: "\<And>var ty vars b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P10 c vars\<rbrakk> \<Longrightarrow> P10 b (VsCons var ty vars)"
+ and a42: "\<And>b. P11 b TvsNil"
+ and a43: "\<And>tvar tkind tvars b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P11 c tvars\<rbrakk>
+ \<Longrightarrow> P11 b (TvsCons tvar tkind tvars)"
+ and a44: "\<And>b. P12 b CvsNil"
+ and a45: "\<And>tvar ckind cvars b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P12 c cvars\<rbrakk>
+ \<Longrightarrow> P12 b (CvsCons tvar ckind cvars)"
+ shows "P1 (a :: 'a :: pt) tkind \<and>
+ P2 (b :: 'b :: pt) ckind \<and>
+ P3 (c :: 'c :: {pt,fs}) ty \<and>
+ P4 (d :: 'd :: pt) ty_lst \<and>
+ P5 (e :: 'e :: {pt,fs}) co \<and>
+ P6 (f :: 'f :: pt) co_lst \<and>
+ P7 (g :: 'g :: {pt,fs}) trm \<and>
+ P8 (h :: 'h :: {pt,fs}) assoc_lst \<and>
+ P9 (i :: 'i :: pt) pat \<and>
+ P10 (j :: 'j :: pt) vars \<and>
+ P11 (k :: 'k :: pt) tvars \<and>
+ P12 (l :: 'l :: pt) cvars"
+proof -
+ have a1: "(\<forall>p a. P1 a (p \<bullet> tkind))" and "(\<forall>p b. P2 b (p \<bullet> ckind))" and "(\<forall>p c. P3 c (p \<bullet> ty))" and "(\<forall>p d. P4 d (p \<bullet> ty_lst))" and "(\<forall>p e. P5 e (p \<bullet> co))" and " (\<forall>p f. P6 f (p \<bullet> co_lst))" and "(\<forall>p g. P7 g (p \<bullet> trm))" and "(\<forall>p h. P8 h (p \<bullet> assoc_lst))" and a1:"(\<forall>p q i. P9 i (permute_bv p (q \<bullet> pat)))" and a2:"(\<forall>p q j. P10 j (permute_bv_vs q (p \<bullet> vars)))" and a3:"(\<forall>p q k. P11 k ( permute_bv_tvs q (p \<bullet> tvars)))" and a4:"(\<forall>p q l. P12 l (permute_bv_cvs q (p \<bullet> cvars)))"
+ apply (induct rule: tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.inducts)
+ apply (tactic {* ALLGOALS (REPEAT o rtac allI) *})
+ apply (tactic {* ALLGOALS (TRY o SOLVED' (simp_tac @{simpset} THEN_ALL_NEW resolve_tac @{thms assms} THEN_ALL_NEW asm_full_simp_tac @{simpset})) *})
+
+(* GOAL1 *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> c \<and>
+ supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> ty)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="TAll (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> ty)"
+ and s="TAll (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> ty)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> ty) - {atom (pa \<bullet> p \<bullet> tvar)}"
+ and s="pa \<bullet> (p \<bullet> supp ty - {p \<bullet> atom tvar})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a08)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+(* GOAL2 *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> cvar))) \<sharp> e \<and>
+ supp (Abs (p \<bullet> {atom cvar}) (p \<bullet> co)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="CAll (p \<bullet> cvar) (p \<bullet> ckind) (p \<bullet> co)"
+ and s="CAll (pa \<bullet> p \<bullet> cvar) (p \<bullet> ckind) (pa \<bullet> p \<bullet> co)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> co) - {atom (pa \<bullet> p \<bullet> cvar)}"
+ and s="pa \<bullet> (p \<bullet> supp co - {p \<bullet> atom cvar})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a15)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+
+(* GOAL3 a copy-and-paste of Goal2 with consts and variable names changed *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> g \<and>
+ supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> trm)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="LAMT (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> trm)"
+ and s="LAMT (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> trm)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> tvar)}"
+ and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom tvar})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a29)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+(* GOAL4 a copy-and-paste *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> cvar))) \<sharp> g \<and>
+ supp (Abs (p \<bullet> {atom cvar}) (p \<bullet> trm)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="LAMC (p \<bullet> cvar) (p \<bullet> ckind) (p \<bullet> trm)"
+ and s="LAMC (pa \<bullet> p \<bullet> cvar) (p \<bullet> ckind) (pa \<bullet> p \<bullet> trm)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> cvar)}"
+ and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom cvar})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a30)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+
+(* GOAL5 a copy-and-paste *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> var))) \<sharp> g \<and>
+ supp (Abs (p \<bullet> {atom var}) (p \<bullet> trm)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="Lam (p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm)"
+ and s="Lam (pa \<bullet> p \<bullet> var) (p \<bullet> ty) (pa \<bullet> p \<bullet> trm)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> var)}"
+ and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom var})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a32)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+
+(* GOAL6 a copy-and-paste *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> var))) \<sharp> g \<and>
+ supp (Abs (p \<bullet> {atom var}) (p \<bullet> trm2)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm)
+ apply(rule_tac t="Let (p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm1) (p \<bullet> trm2)"
+ and s="Let (pa \<bullet> p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm1) (pa \<bullet> p \<bullet> trm2)" in subst)
+ apply (simp add: eq_iff)
+ apply (rule_tac x="-pa" in exI)
+ apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
+ apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm2) - {atom (pa \<bullet> p \<bullet> var)}"
+ and s="pa \<bullet> (p \<bullet> supp trm2 - {p \<bullet> atom var})" in subst)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts[symmetric])
+ apply (rule conjI)
+ apply (rule supp_perm_eq)
+ apply (simp add: eqvts)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply (simp add: eqvts)
+ apply (subst supp_perm_eq)
+ apply (subst supp_finite_atom_set)
+ apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
+ apply (simp add: eqvts)
+ apply assumption
+ apply (simp add: fresh_star_minus_perm)
+ apply (rule a34)
+ apply simp
+ apply simp
+ apply(rotate_tac 2)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2_atom)
+ apply (simp add: finite_supp)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_def)
+ apply (simp only: supp_abs eqvts)
+ apply blast
+
+(* MAIN ACons Goal *)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (set (bv (p \<bullet> pat)))) \<sharp>* h \<and>
+ supp (Abs_lst (p \<bullet> (bv pat)) (p \<bullet> trm)) \<sharp>* pa)")
+ apply clarify
+ apply (simp only: perm eqvts)
+ apply (subst ACons_subst)
+ apply assumption
+ apply (rule a38)
+ apply simp
+ apply(rotate_tac 1)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply simp
+ apply (simp add: perm_bv2[symmetric])
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2)
+ apply (simp add: fin_bv)
+ apply (simp add: finite_supp)
+ apply (simp add: supp_abs)
+ apply (simp add: finite_supp)
+ apply (simp add: fresh_star_def fresh_def supp_abs eqvts)
+ done
+ then have b: "P1 a (0 \<bullet> tkind)" and "P2 b (0 \<bullet> ckind)" "P3 c (0 \<bullet> ty)" and "P4 d (0 \<bullet> ty_lst)" and "P5 e (0 \<bullet> co)" and "P6 f (0 \<bullet> co_lst)" and "P7 g (0 \<bullet> trm)" and "P8 h (0 \<bullet> assoc_lst)" by (blast+)
+ moreover have "P9 i (permute_bv 0 (0 \<bullet> pat))" and "P10 j (permute_bv_vs 0 (0 \<bullet> vars))" and "P11 k (permute_bv_tvs 0 (0 \<bullet> tvars))" and "P12 l (permute_bv_cvs 0 (0 \<bullet> cvars))" using a1 a2 a3 a4 by (blast+)
+ ultimately show ?thesis by (simp_all add: permute_bv_zero1 permute_bv_zero2)
+qed
+
+section {* test about equivariance for alpha *}
+
+(* this should not be an equivariance rule *)
+(* for the moment, we force it to be *)
+
+(*declare permute_pure[eqvt]*)
+(*setup {* Context.theory_map (Nominal_ThmDecls.add_thm @{thm "permute_pure"}) *)
+
+thm eqvts
+thm eqvts_raw
+
+declare permute_tkind_raw_permute_ckind_raw_permute_ty_raw_permute_ty_lst_raw_permute_co_raw_permute_co_lst_raw_permute_trm_raw_permute_assoc_lst_raw_permute_pat_raw_permute_vars_raw_permute_tvars_raw_permute_cvars_raw.simps[eqvt]
+declare alpha_gen_eqvt[eqvt]
+
+equivariance alpha_tkind_raw
+
+thm eqvts
+thm eqvts_raw
+
+end
+
--- a/Nominal/Ex/ExCoreHaskell.thy Sun May 09 12:26:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,675 +0,0 @@
-theory ExCoreHaskell
-imports "../NewParser"
-begin
-
-(* core haskell *)
-
-atom_decl var
-atom_decl cvar
-atom_decl tvar
-
-(* there are types, coercion types and regular types list-data-structure *)
-
-nominal_datatype tkind =
- KStar
-| KFun "tkind" "tkind"
-and ckind =
- CKEq "ty" "ty"
-and ty =
- TVar "tvar"
-| TC "string"
-| TApp "ty" "ty"
-| TFun "string" "ty_lst"
-| TAll tv::"tvar" "tkind" T::"ty" bind_set tv in T
-| TEq "ckind" "ty"
-and ty_lst =
- TsNil
-| TsCons "ty" "ty_lst"
-and co =
- CVar "cvar"
-| CConst "string"
-| CApp "co" "co"
-| CFun "string" "co_lst"
-| CAll cv::"cvar" "ckind" C::"co" bind_set cv in C
-| CEq "ckind" "co"
-| CRefl "ty"
-| CSym "co"
-| CCir "co" "co"
-| CAt "co" "ty"
-| CLeft "co"
-| CRight "co"
-| CSim "co" "co"
-| CRightc "co"
-| CLeftc "co"
-| CCoe "co" "co"
-and co_lst =
- CsNil
-| CsCons "co" "co_lst"
-and trm =
- Var "var"
-| K "string"
-| LAMT tv::"tvar" "tkind" t::"trm" bind_set tv in t
-| LAMC cv::"cvar" "ckind" t::"trm" bind_set cv in t
-| AppT "trm" "ty"
-| AppC "trm" "co"
-| Lam v::"var" "ty" t::"trm" bind_set v in t
-| App "trm" "trm"
-| Let x::"var" "ty" "trm" t::"trm" bind_set x in t
-| Case "trm" "assoc_lst"
-| Cast "trm" "ty" --"ty is supposed to be a coercion type only"
-and assoc_lst =
- ANil
-| ACons p::"pat" t::"trm" "assoc_lst" bind "bv p" in t
-and pat =
- Kpat "string" "tvars" "cvars" "vars"
-and vars =
- VsNil
-| VsCons "var" "ty" "vars"
-and tvars =
- TvsNil
-| TvsCons "tvar" "tkind" "tvars"
-and cvars =
- CvsNil
-| CvsCons "cvar" "ckind" "cvars"
-binder
- bv :: "pat \<Rightarrow> atom list"
-and bv_vs :: "vars \<Rightarrow> atom list"
-and bv_tvs :: "tvars \<Rightarrow> atom list"
-and bv_cvs :: "cvars \<Rightarrow> atom list"
-where
- "bv (Kpat s tvts tvcs vs) = append (bv_tvs tvts) (append (bv_cvs tvcs) (bv_vs vs))"
-| "bv_vs VsNil = []"
-| "bv_vs (VsCons v k t) = (atom v) # bv_vs t"
-| "bv_tvs TvsNil = []"
-| "bv_tvs (TvsCons v k t) = (atom v) # bv_tvs t"
-| "bv_cvs CvsNil = []"
-| "bv_cvs (CvsCons v k t) = (atom v) # bv_cvs t"
-
-lemmas fv_supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.supp(1-9,11,13,15)
-lemmas supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.fv[simplified fv_supp]
-lemmas perm=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.perm
-lemmas eq_iff=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.eq_iff
-lemmas inducts=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.inducts
-
-lemmas alpha_inducts=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vars_raw_alpha_tvars_raw_alpha_cvars_raw_alpha_bv_raw_alpha_bv_vs_raw_alpha_bv_tvs_raw_alpha_bv_cvs_raw.inducts
-lemmas alpha_intros=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vars_raw_alpha_tvars_raw_alpha_cvars_raw_alpha_bv_raw_alpha_bv_vs_raw_alpha_bv_tvs_raw_alpha_bv_cvs_raw.intros
-
-lemma fresh_star_minus_perm: "as \<sharp>* - p = as \<sharp>* (p :: perm)"
- unfolding fresh_star_def Ball_def
- by auto (simp_all add: fresh_minus_perm)
-
-primrec permute_bv_vs_raw
-where "permute_bv_vs_raw p VsNil_raw = VsNil_raw"
-| "permute_bv_vs_raw p (VsCons_raw v t l) = VsCons_raw (p \<bullet> v) t (permute_bv_vs_raw p l)"
-primrec permute_bv_cvs_raw
-where "permute_bv_cvs_raw p CvsNil_raw = CvsNil_raw"
-| "permute_bv_cvs_raw p (CvsCons_raw v t l) = CvsCons_raw (p \<bullet> v) t (permute_bv_cvs_raw p l)"
-primrec permute_bv_tvs_raw
-where "permute_bv_tvs_raw p TvsNil_raw = TvsNil_raw"
-| "permute_bv_tvs_raw p (TvsCons_raw v t l) = TvsCons_raw (p \<bullet> v) t (permute_bv_tvs_raw p l)"
-primrec permute_bv_raw
-where "permute_bv_raw p (Kpat_raw c l1 l2 l3) =
- Kpat_raw c (permute_bv_tvs_raw p l1) (permute_bv_cvs_raw p l2) (permute_bv_vs_raw p l3)"
-
-quotient_definition "permute_bv_vs :: perm \<Rightarrow> vars \<Rightarrow> vars"
-is "permute_bv_vs_raw"
-quotient_definition "permute_bv_cvs :: perm \<Rightarrow> cvars \<Rightarrow> cvars"
-is "permute_bv_cvs_raw"
-quotient_definition "permute_bv_tvs :: perm \<Rightarrow> tvars \<Rightarrow> tvars"
-is "permute_bv_tvs_raw"
-quotient_definition "permute_bv :: perm \<Rightarrow> pat \<Rightarrow> pat"
-is "permute_bv_raw"
-
-lemma rsp_pre:
- "alpha_tvars_raw d a \<Longrightarrow> alpha_tvars_raw (permute_bv_tvs_raw x d) (permute_bv_tvs_raw x a)"
- "alpha_cvars_raw e b \<Longrightarrow> alpha_cvars_raw (permute_bv_cvs_raw x e) (permute_bv_cvs_raw x b)"
- "alpha_vars_raw f c \<Longrightarrow> alpha_vars_raw (permute_bv_vs_raw x f) (permute_bv_vs_raw x c)"
- apply (erule_tac [!] alpha_inducts)
- apply (simp_all only: alpha_intros perm permute_bv_tvs_raw.simps permute_bv_cvs_raw.simps permute_bv_vs_raw.simps)
- done
-
-lemma [quot_respect]:
- "(op = ===> alpha_pat_raw ===> alpha_pat_raw) permute_bv_raw permute_bv_raw"
- "(op = ===> alpha_tvars_raw ===> alpha_tvars_raw) permute_bv_tvs_raw permute_bv_tvs_raw"
- "(op = ===> alpha_cvars_raw ===> alpha_cvars_raw) permute_bv_cvs_raw permute_bv_cvs_raw"
- "(op = ===> alpha_vars_raw ===> alpha_vars_raw) permute_bv_vs_raw permute_bv_vs_raw"
- apply (simp_all add: rsp_pre)
- apply clarify
- apply (erule_tac alpha_inducts)
- apply (simp_all)
- apply (rule alpha_intros)
- apply (simp_all add: rsp_pre)
- done
-
-thm permute_bv_raw.simps[no_vars]
- permute_bv_vs_raw.simps[quot_lifted]
- permute_bv_cvs_raw.simps[quot_lifted]
- permute_bv_tvs_raw.simps[quot_lifted]
-
-lemma permute_bv_pre:
- "permute_bv p (Kpat c l1 l2 l3) =
- Kpat c (permute_bv_tvs p l1) (permute_bv_cvs p l2) (permute_bv_vs p l3)"
- by (lifting permute_bv_raw.simps)
-
-lemmas permute_bv[simp] =
- permute_bv_pre
- permute_bv_vs_raw.simps[quot_lifted]
- permute_bv_cvs_raw.simps[quot_lifted]
- permute_bv_tvs_raw.simps[quot_lifted]
-
-lemma perm_bv1:
- "p \<bullet> bv_cvs b = bv_cvs (permute_bv_cvs p b)"
- "p \<bullet> bv_tvs c = bv_tvs (permute_bv_tvs p c)"
- "p \<bullet> bv_vs d = bv_vs (permute_bv_vs p d)"
- apply(induct b rule: inducts(12))
- apply(simp_all add:permute_bv eqvts)
- apply(induct c rule: inducts(11))
- apply(simp_all add:permute_bv eqvts)
- apply(induct d rule: inducts(10))
- apply(simp_all add:permute_bv eqvts)
- done
-
-lemma perm_bv2:
- "p \<bullet> bv l = bv (permute_bv p l)"
- apply(induct l rule: inducts(9))
- apply(simp_all add:permute_bv)
- apply(simp add: perm_bv1[symmetric])
- apply(simp add: eqvts)
- done
-
-lemma alpha_perm_bn1:
- "alpha_bv_tvs tvars (permute_bv_tvs q tvars)"
- "alpha_bv_cvs cvars (permute_bv_cvs q cvars)"
- "alpha_bv_vs vars (permute_bv_vs q vars)"
- apply(induct tvars rule: inducts(11))
- apply(simp_all add:permute_bv eqvts eq_iff)
- apply(induct cvars rule: inducts(12))
- apply(simp_all add:permute_bv eqvts eq_iff)
- apply(induct vars rule: inducts(10))
- apply(simp_all add:permute_bv eqvts eq_iff)
- done
-
-lemma alpha_perm_bn:
- "alpha_bv pat (permute_bv q pat)"
- apply(induct pat rule: inducts(9))
- apply(simp_all add:permute_bv eqvts eq_iff alpha_perm_bn1)
- done
-
-lemma ACons_subst:
- "supp (Abs_lst (bv pat) trm) \<sharp>* q \<Longrightarrow> (ACons pat trm al) = ACons (permute_bv q pat) (q \<bullet> trm) al"
- apply (simp only: eq_iff)
- apply (simp add: alpha_perm_bn)
- apply (rule_tac x="q" in exI)
- apply (simp add: alphas)
- apply (simp add: perm_bv2[symmetric])
- apply (simp add: eqvts[symmetric])
- apply (simp add: supp_abs)
- apply (simp add: fv_supp)
- apply (rule supp_perm_eq[symmetric])
- apply (subst supp_finite_atom_set)
- apply (rule finite_Diff)
- apply (simp add: finite_supp)
- apply (assumption)
- done
-
-lemma permute_bv_zero1:
- "permute_bv_cvs 0 b = b"
- "permute_bv_tvs 0 c = c"
- "permute_bv_vs 0 d = d"
- apply(induct b rule: inducts(12))
- apply(simp_all add:permute_bv eqvts)
- apply(induct c rule: inducts(11))
- apply(simp_all add:permute_bv eqvts)
- apply(induct d rule: inducts(10))
- apply(simp_all add:permute_bv eqvts)
- done
-
-lemma permute_bv_zero2:
- "permute_bv 0 a = a"
- apply(induct a rule: inducts(9))
- apply(simp_all add:permute_bv eqvts permute_bv_zero1)
- done
-
-lemma fv_alpha1: "fv_bv_tvs x \<sharp>* pa \<Longrightarrow> alpha_bv_tvs (pa \<bullet> x) x"
- apply (induct x rule: inducts(11))
- apply (simp_all add: eq_iff fresh_star_union)
- done
-
-lemma fv_alpha2: "fv_bv_cvs x \<sharp>* pa \<Longrightarrow> alpha_bv_cvs (pa \<bullet> x) x"
-apply (induct x rule: inducts(12))
-apply (rule TrueI)+
-apply (simp_all add: eq_iff fresh_star_union)
-apply (subst supp_perm_eq)
-apply (simp_all add: fv_supp)
-done
-
-lemma fv_alpha3: "fv_bv_vs x \<sharp>* pa \<Longrightarrow> alpha_bv_vs (pa \<bullet> x) x"
-apply (induct x rule: inducts(10))
-apply (rule TrueI)+
-apply (simp_all add: fresh_star_union eq_iff)
-apply (subst supp_perm_eq)
-apply (simp_all add: fv_supp)
-done
-
-lemma fv_alpha: "fv_bv x \<sharp>* pa \<Longrightarrow> alpha_bv (pa \<bullet> x) x"
-apply (induct x rule: inducts(9))
-apply (rule TrueI)+
-apply (simp_all add: eq_iff fresh_star_union)
-apply (simp add: fv_alpha1 fv_alpha2 fv_alpha3)
-apply (subst supp_perm_eq)
-apply (simp_all add: fv_supp)
-done
-
-lemma fin1: "finite (fv_bv_tvs x)"
-apply (induct x rule: inducts(11))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma fin2: "finite (fv_bv_cvs x)"
-apply (induct x rule: inducts(12))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma fin3: "finite (fv_bv_vs x)"
-apply (induct x rule: inducts(10))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma fin_fv_bv: "finite (fv_bv x)"
-apply (induct x rule: inducts(9))
-apply (rule TrueI)+
-defer
-apply (rule TrueI)+
-apply (simp add: fin1 fin2 fin3)
-apply (rule finite_supp)
-done
-
-lemma finb1: "finite (set (bv_tvs x))"
-apply (induct x rule: inducts(11))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma finb2: "finite (set (bv_cvs x))"
-apply (induct x rule: inducts(12))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma finb3: "finite (set (bv_vs x))"
-apply (induct x rule: inducts(10))
-apply (simp_all add: fv_supp finite_supp)
-done
-
-lemma fin_bv: "finite (set (bv x))"
-apply (induct x rule: inducts(9))
-apply (simp_all add: finb1 finb2 finb3)
-done
-
-lemma strong_induction_principle:
- assumes a01: "\<And>b. P1 b KStar"
- and a02: "\<And>tkind1 tkind2 b. \<lbrakk>\<And>c. P1 c tkind1; \<And>c. P1 c tkind2\<rbrakk> \<Longrightarrow> P1 b (KFun tkind1 tkind2)"
- and a03: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P2 b (CKEq ty1 ty2)"
- and a04: "\<And>tvar b. P3 b (TVar tvar)"
- and a05: "\<And>string b. P3 b (TC string)"
- and a06: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P3 b (TApp ty1 ty2)"
- and a07: "\<And>string ty_lst b. \<lbrakk>\<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P3 b (TFun string ty_lst)"
- and a08: "\<And>tvar tkind ty b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P3 c ty; atom tvar \<sharp> b\<rbrakk>
- \<Longrightarrow> P3 b (TAll tvar tkind ty)"
- and a09: "\<And>ck ty b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P3 b (TEq ck ty)"
- and a10: "\<And>b. P4 b TsNil"
- and a11: "\<And>ty ty_lst b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P4 b (TsCons ty ty_lst)"
- and a12: "\<And>string b. P5 b (CVar string)"
- and a12a:"\<And>str b. P5 b (CConst str)"
- and a13: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CApp co1 co2)"
- and a14: "\<And>string co_lst b. \<lbrakk>\<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P5 b (CFun string co_lst)"
- and a15: "\<And>tvar ckind co b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P5 c co; atom tvar \<sharp> b\<rbrakk>
- \<Longrightarrow> P5 b (CAll tvar ckind co)"
- and a16: "\<And>ck co b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CEq ck co)"
- and a17: "\<And>ty b. \<lbrakk>\<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CRefl ty)"
- and a17a: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CSym co)"
- and a18: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCir co1 co2)"
- and a18a:"\<And>co ty b. \<lbrakk>\<And>c. P5 c co; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CAt co ty)"
- and a19: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeft co)"
- and a20: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRight co)"
- and a21: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CSim co1 co2)"
- and a22: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRightc co)"
- and a23: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeftc co)"
- and a24: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCoe co1 co2)"
- and a25: "\<And>b. P6 b CsNil"
- and a26: "\<And>co co_lst b. \<lbrakk>\<And>c. P5 c co; \<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P6 b (CsCons co co_lst)"
- and a27: "\<And>var b. P7 b (Var var)"
- and a28: "\<And>string b. P7 b (K string)"
- and a29: "\<And>tvar tkind trm b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>
- \<Longrightarrow> P7 b (LAMT tvar tkind trm)"
- and a30: "\<And>tvar ckind trm b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>
- \<Longrightarrow> P7 b (LAMC tvar ckind trm)"
- and a31: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (AppT trm ty)"
- and a31a:"\<And>trm co b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P7 b (AppC trm co)"
- and a32: "\<And>var ty trm b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm; atom var \<sharp> b\<rbrakk> \<Longrightarrow> P7 b (Lam var ty trm)"
- and a33: "\<And>trm1 trm2 b. \<lbrakk>\<And>c. P7 c trm1; \<And>c. P7 c trm2\<rbrakk> \<Longrightarrow> P7 b (App trm1 trm2)"
- and a34: "\<And>var ty trm1 trm2 b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm1; \<And>c. P7 c trm2; atom var \<sharp> b\<rbrakk>
- \<Longrightarrow> P7 b (Let var ty trm1 trm2)"
- and a35: "\<And>trm assoc_lst b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P8 c assoc_lst\<rbrakk> \<Longrightarrow> P7 b (Case trm assoc_lst)"
- and a36: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (Cast trm ty)"
- and a37: "\<And>b. P8 b ANil"
- and a38: "\<And>pat trm assoc_lst b. \<lbrakk>\<And>c. P9 c pat; \<And>c. P7 c trm; \<And>c. P8 c assoc_lst; set (bv (pat)) \<sharp>* b\<rbrakk>
- \<Longrightarrow> P8 b (ACons pat trm assoc_lst)"
- and a39: "\<And>string tvars cvars vars b. \<lbrakk>\<And>c. P11 c tvars; \<And>c. P12 c cvars; \<And>c. P10 c vars\<rbrakk>
- \<Longrightarrow> P9 b (Kpat string tvars cvars vars)"
- and a40: "\<And>b. P10 b VsNil"
- and a41: "\<And>var ty vars b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P10 c vars\<rbrakk> \<Longrightarrow> P10 b (VsCons var ty vars)"
- and a42: "\<And>b. P11 b TvsNil"
- and a43: "\<And>tvar tkind tvars b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P11 c tvars\<rbrakk>
- \<Longrightarrow> P11 b (TvsCons tvar tkind tvars)"
- and a44: "\<And>b. P12 b CvsNil"
- and a45: "\<And>tvar ckind cvars b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P12 c cvars\<rbrakk>
- \<Longrightarrow> P12 b (CvsCons tvar ckind cvars)"
- shows "P1 (a :: 'a :: pt) tkind \<and>
- P2 (b :: 'b :: pt) ckind \<and>
- P3 (c :: 'c :: {pt,fs}) ty \<and>
- P4 (d :: 'd :: pt) ty_lst \<and>
- P5 (e :: 'e :: {pt,fs}) co \<and>
- P6 (f :: 'f :: pt) co_lst \<and>
- P7 (g :: 'g :: {pt,fs}) trm \<and>
- P8 (h :: 'h :: {pt,fs}) assoc_lst \<and>
- P9 (i :: 'i :: pt) pat \<and>
- P10 (j :: 'j :: pt) vars \<and>
- P11 (k :: 'k :: pt) tvars \<and>
- P12 (l :: 'l :: pt) cvars"
-proof -
- have a1: "(\<forall>p a. P1 a (p \<bullet> tkind))" and "(\<forall>p b. P2 b (p \<bullet> ckind))" and "(\<forall>p c. P3 c (p \<bullet> ty))" and "(\<forall>p d. P4 d (p \<bullet> ty_lst))" and "(\<forall>p e. P5 e (p \<bullet> co))" and " (\<forall>p f. P6 f (p \<bullet> co_lst))" and "(\<forall>p g. P7 g (p \<bullet> trm))" and "(\<forall>p h. P8 h (p \<bullet> assoc_lst))" and a1:"(\<forall>p q i. P9 i (permute_bv p (q \<bullet> pat)))" and a2:"(\<forall>p q j. P10 j (permute_bv_vs q (p \<bullet> vars)))" and a3:"(\<forall>p q k. P11 k ( permute_bv_tvs q (p \<bullet> tvars)))" and a4:"(\<forall>p q l. P12 l (permute_bv_cvs q (p \<bullet> cvars)))"
- apply (induct rule: tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vars_tvars_cvars.inducts)
- apply (tactic {* ALLGOALS (REPEAT o rtac allI) *})
- apply (tactic {* ALLGOALS (TRY o SOLVED' (simp_tac @{simpset} THEN_ALL_NEW resolve_tac @{thms assms} THEN_ALL_NEW asm_full_simp_tac @{simpset})) *})
-
-(* GOAL1 *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> c \<and>
- supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> ty)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="TAll (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> ty)"
- and s="TAll (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> ty)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> ty) - {atom (pa \<bullet> p \<bullet> tvar)}"
- and s="pa \<bullet> (p \<bullet> supp ty - {p \<bullet> atom tvar})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a08)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-(* GOAL2 *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> cvar))) \<sharp> e \<and>
- supp (Abs (p \<bullet> {atom cvar}) (p \<bullet> co)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="CAll (p \<bullet> cvar) (p \<bullet> ckind) (p \<bullet> co)"
- and s="CAll (pa \<bullet> p \<bullet> cvar) (p \<bullet> ckind) (pa \<bullet> p \<bullet> co)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> co) - {atom (pa \<bullet> p \<bullet> cvar)}"
- and s="pa \<bullet> (p \<bullet> supp co - {p \<bullet> atom cvar})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a15)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-
-(* GOAL3 a copy-and-paste of Goal2 with consts and variable names changed *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> g \<and>
- supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> trm)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="LAMT (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> trm)"
- and s="LAMT (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> trm)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> tvar)}"
- and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom tvar})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a29)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-(* GOAL4 a copy-and-paste *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> cvar))) \<sharp> g \<and>
- supp (Abs (p \<bullet> {atom cvar}) (p \<bullet> trm)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="LAMC (p \<bullet> cvar) (p \<bullet> ckind) (p \<bullet> trm)"
- and s="LAMC (pa \<bullet> p \<bullet> cvar) (p \<bullet> ckind) (pa \<bullet> p \<bullet> trm)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> cvar)}"
- and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom cvar})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a30)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-
-(* GOAL5 a copy-and-paste *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> var))) \<sharp> g \<and>
- supp (Abs (p \<bullet> {atom var}) (p \<bullet> trm)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="Lam (p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm)"
- and s="Lam (pa \<bullet> p \<bullet> var) (p \<bullet> ty) (pa \<bullet> p \<bullet> trm)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> var)}"
- and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom var})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a32)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-
-(* GOAL6 a copy-and-paste *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> var))) \<sharp> g \<and>
- supp (Abs (p \<bullet> {atom var}) (p \<bullet> trm2)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm)
- apply(rule_tac t="Let (p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm1) (p \<bullet> trm2)"
- and s="Let (pa \<bullet> p \<bullet> var) (p \<bullet> ty) (p \<bullet> trm1) (pa \<bullet> p \<bullet> trm2)" in subst)
- apply (simp add: eq_iff)
- apply (rule_tac x="-pa" in exI)
- apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)
- apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm2) - {atom (pa \<bullet> p \<bullet> var)}"
- and s="pa \<bullet> (p \<bullet> supp trm2 - {p \<bullet> atom var})" in subst)
- apply (simp add: eqvts)
- apply (simp add: eqvts[symmetric])
- apply (rule conjI)
- apply (rule supp_perm_eq)
- apply (simp add: eqvts)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply (simp add: eqvts)
- apply (subst supp_perm_eq)
- apply (subst supp_finite_atom_set)
- apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)
- apply (simp add: eqvts)
- apply assumption
- apply (simp add: fresh_star_minus_perm)
- apply (rule a34)
- apply simp
- apply simp
- apply(rotate_tac 2)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2_atom)
- apply (simp add: finite_supp)
- apply (simp add: finite_supp)
- apply (simp add: fresh_def)
- apply (simp only: supp_abs eqvts)
- apply blast
-
-(* MAIN ACons Goal *)
- apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (set (bv (p \<bullet> pat)))) \<sharp>* h \<and>
- supp (Abs_lst (p \<bullet> (bv pat)) (p \<bullet> trm)) \<sharp>* pa)")
- apply clarify
- apply (simp only: perm eqvts)
- apply (subst ACons_subst)
- apply assumption
- apply (rule a38)
- apply simp
- apply(rotate_tac 1)
- apply(erule_tac x="(pa + p)" in allE)
- apply simp
- apply simp
- apply (simp add: perm_bv2[symmetric])
- apply (simp add: eqvts eqvts_raw)
- apply (rule at_set_avoiding2)
- apply (simp add: fin_bv)
- apply (simp add: finite_supp)
- apply (simp add: supp_abs)
- apply (simp add: finite_supp)
- apply (simp add: fresh_star_def fresh_def supp_abs eqvts)
- done
- then have b: "P1 a (0 \<bullet> tkind)" and "P2 b (0 \<bullet> ckind)" "P3 c (0 \<bullet> ty)" and "P4 d (0 \<bullet> ty_lst)" and "P5 e (0 \<bullet> co)" and "P6 f (0 \<bullet> co_lst)" and "P7 g (0 \<bullet> trm)" and "P8 h (0 \<bullet> assoc_lst)" by (blast+)
- moreover have "P9 i (permute_bv 0 (0 \<bullet> pat))" and "P10 j (permute_bv_vs 0 (0 \<bullet> vars))" and "P11 k (permute_bv_tvs 0 (0 \<bullet> tvars))" and "P12 l (permute_bv_cvs 0 (0 \<bullet> cvars))" using a1 a2 a3 a4 by (blast+)
- ultimately show ?thesis by (simp_all add: permute_bv_zero1 permute_bv_zero2)
-qed
-
-section {* test about equivariance for alpha *}
-
-(* this should not be an equivariance rule *)
-(* for the moment, we force it to be *)
-
-(*declare permute_pure[eqvt]*)
-(*setup {* Context.theory_map (Nominal_ThmDecls.add_thm @{thm "permute_pure"}) *)
-
-thm eqvts
-thm eqvts_raw
-
-declare permute_tkind_raw_permute_ckind_raw_permute_ty_raw_permute_ty_lst_raw_permute_co_raw_permute_co_lst_raw_permute_trm_raw_permute_assoc_lst_raw_permute_pat_raw_permute_vars_raw_permute_tvars_raw_permute_cvars_raw.simps[eqvt]
-declare alpha_gen_eqvt[eqvt]
-
-equivariance alpha_tkind_raw
-
-thm eqvts
-thm eqvts_raw
-
-end
-
--- a/Nominal/Ex/ExLF.thy Sun May 09 12:26:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,35 +0,0 @@
-theory ExLF
-imports "../NewParser"
-begin
-
-atom_decl name
-atom_decl ident
-
-nominal_datatype kind =
- Type
- | KPi "ty" n::"name" k::"kind" bind_set n in k
-and ty =
- TConst "ident"
- | TApp "ty" "trm"
- | TPi "ty" n::"name" t::"ty" bind_set n in t
-and trm =
- Const "ident"
- | Var "name"
- | App "trm" "trm"
- | Lam "ty" n::"name" t::"trm" bind_set n in t
-
-thm kind_ty_trm.supp
-
-declare permute_kind_raw_permute_ty_raw_permute_trm_raw.simps[eqvt]
-declare alpha_gen_eqvt[eqvt]
-
-equivariance alpha_trm_raw
-
-
-
-
-end
-
-
-
-
--- a/Nominal/Ex/ExLeroy.thy Sun May 09 12:26:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,79 +0,0 @@
-theory ExLeroy
-imports "../NewParser"
-begin
-
-(* example form Leroy 96 about modules; OTT *)
-
-atom_decl name
-
-nominal_datatype mexp =
- Acc "path"
-| Stru "body"
-| Funct x::"name" "sexp" m::"mexp" bind_set x in m
-| FApp "mexp" "path"
-| Ascr "mexp" "sexp"
-and body =
- Empty
-| Seq c::defn d::"body" bind_set "cbinders c" in d
-and defn =
- Type "name" "ty"
-| Dty "name"
-| DStru "name" "mexp"
-| Val "name" "trm"
-and sexp =
- Sig sbody
-| SFunc "name" "sexp" "sexp"
-and sbody =
- SEmpty
-| SSeq C::spec D::sbody bind_set "Cbinders C" in D
-and spec =
- Type1 "name"
-| Type2 "name" "ty"
-| SStru "name" "sexp"
-| SVal "name" "ty"
-and ty =
- Tyref1 "name"
-| Tyref2 "path" "ty"
-| Fun "ty" "ty"
-and path =
- Sref1 "name"
-| Sref2 "path" "name"
-and trm =
- Tref1 "name"
-| Tref2 "path" "name"
-| Lam' v::"name" "ty" M::"trm" bind_set v in M
-| App' "trm" "trm"
-| Let' "body" "trm"
-binder
- cbinders :: "defn \<Rightarrow> atom set"
-and Cbinders :: "spec \<Rightarrow> atom set"
-where
- "cbinders (Type t T) = {atom t}"
-| "cbinders (Dty t) = {atom t}"
-| "cbinders (DStru x s) = {atom x}"
-| "cbinders (Val v M) = {atom v}"
-| "Cbinders (Type1 t) = {atom t}"
-| "Cbinders (Type2 t T) = {atom t}"
-| "Cbinders (SStru x S) = {atom x}"
-| "Cbinders (SVal v T) = {atom v}"
-
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.fv
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.eq_iff
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.bn
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.perm
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.induct
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.inducts
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.distinct
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.supp(1-3,5-7,9-10)
-thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.fv[simplified mexp_body_defn_sexp_sbody_spec_ty_path_trm.supp(1-3,5-7,9-10)]
-
-declare permute_mexp_raw_permute_body_raw_permute_defn_raw_permute_sexp_raw_permute_sbody_raw_permute_spec_raw_permute_ty_raw_permute_path_raw_permute_trm_raw.simps[eqvt]
-declare alpha_gen_eqvt[eqvt]
-
-equivariance alpha_trm_raw
-
-
-end
-
-
-
--- a/Nominal/Ex/ExNotRsp.thy Sun May 09 12:26:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,54 +0,0 @@
-theory ExNotRsp
-imports "../NewParser"
-begin
-
-atom_decl name
-
-(* this example binds bound names and
- so is not respectful *)
-(*
-nominal_datatype trm =
- Vr "name"
-| Lm x::"name" t::"trm" bind x in t
-| Lt left::"trm" right::"trm" bind "bv left" in right
-binder
- bv
-where
- "bv (Vr n) = {}"
-| "bv (Lm n t) = {atom n} \<union> bv t"
-| "bv (Lt l r) = bv l \<union> bv r"
-*)
-
-(* this example uses "-" in the binding function;
- at the moment this is unsupported *)
-(*
-nominal_datatype trm' =
- Vr' "name"
-| Lm' l::"name" r::"trm'" bind l in r
-| Lt' l::"trm'" r::"trm'" bind "bv' l" in r
-binder
- bv'
-where
- "bv' (Vr' n) = {atom n}"
-| "bv' (Lm' n t) = bv' t - {atom n}"
-| "bv' (Lt' l r) = bv' l \<union> bv' r"
-*)
-
-(* this example again binds bound names *)
-(*
-nominal_datatype trm'' =
- Va'' "name"
-| Lm'' n::"name" l::"trm''" bind n in l
-and bla'' =
- Bla'' f::"trm''" s::"trm''" bind "bv'' f" in s
-binder
- bv''
-where
- "bv'' (Vm'' x) = {}"
-| "bv'' (Lm'' x b) = {atom x}"
-*)
-
-end
-
-
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/LF.thy Sun May 09 12:38:59 2010 +0100
@@ -0,0 +1,35 @@
+theory LF
+imports "../NewParser"
+begin
+
+atom_decl name
+atom_decl ident
+
+nominal_datatype kind =
+ Type
+ | KPi "ty" n::"name" k::"kind" bind n in k
+and ty =
+ TConst "ident"
+ | TApp "ty" "trm"
+ | TPi "ty" n::"name" t::"ty" bind n in t
+and trm =
+ Const "ident"
+ | Var "name"
+ | App "trm" "trm"
+ | Lam "ty" n::"name" t::"trm" bind n in t
+
+thm kind_ty_trm.supp
+
+declare permute_kind_raw_permute_ty_raw_permute_trm_raw.simps[eqvt]
+declare alpha_gen_eqvt[eqvt]
+
+equivariance alpha_trm_raw
+
+
+
+
+end
+
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/Modules.thy Sun May 09 12:38:59 2010 +0100
@@ -0,0 +1,80 @@
+theory Modules
+imports "../NewParser"
+begin
+
+(* example from Leroy'96 about modules;
+ see OTT example by Owens *)
+
+atom_decl name
+
+nominal_datatype mexp =
+ Acc "path"
+| Stru "body"
+| Funct x::"name" "sexp" m::"mexp" bind_set x in m
+| FApp "mexp" "path"
+| Ascr "mexp" "sexp"
+and body =
+ Empty
+| Seq c::defn d::"body" bind_set "cbinders c" in d
+and defn =
+ Type "name" "ty"
+| Dty "name"
+| DStru "name" "mexp"
+| Val "name" "trm"
+and sexp =
+ Sig sbody
+| SFunc "name" "sexp" "sexp"
+and sbody =
+ SEmpty
+| SSeq C::spec D::sbody bind_set "Cbinders C" in D
+and spec =
+ Type1 "name"
+| Type2 "name" "ty"
+| SStru "name" "sexp"
+| SVal "name" "ty"
+and ty =
+ Tyref1 "name"
+| Tyref2 "path" "ty"
+| Fun "ty" "ty"
+and path =
+ Sref1 "name"
+| Sref2 "path" "name"
+and trm =
+ Tref1 "name"
+| Tref2 "path" "name"
+| Lam' v::"name" "ty" M::"trm" bind_set v in M
+| App' "trm" "trm"
+| Let' "body" "trm"
+binder
+ cbinders :: "defn \<Rightarrow> atom set"
+and Cbinders :: "spec \<Rightarrow> atom set"
+where
+ "cbinders (Type t T) = {atom t}"
+| "cbinders (Dty t) = {atom t}"
+| "cbinders (DStru x s) = {atom x}"
+| "cbinders (Val v M) = {atom v}"
+| "Cbinders (Type1 t) = {atom t}"
+| "Cbinders (Type2 t T) = {atom t}"
+| "Cbinders (SStru x S) = {atom x}"
+| "Cbinders (SVal v T) = {atom v}"
+
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.fv
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.eq_iff
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.bn
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.perm
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.induct
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.inducts
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.distinct
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.supp(1-3,5-7,9-10)
+thm mexp_body_defn_sexp_sbody_spec_ty_path_trm.fv[simplified mexp_body_defn_sexp_sbody_spec_ty_path_trm.supp(1-3,5-7,9-10)]
+
+declare permute_mexp_raw_permute_body_raw_permute_defn_raw_permute_sexp_raw_permute_sbody_raw_permute_spec_raw_permute_ty_raw_permute_path_raw_permute_trm_raw.simps[eqvt]
+declare alpha_gen_eqvt[eqvt]
+
+equivariance alpha_trm_raw
+
+
+end
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/NoneExamples.thy Sun May 09 12:38:59 2010 +0100
@@ -0,0 +1,54 @@
+theory NoneExamples
+imports "../NewParser"
+begin
+
+atom_decl name
+
+(* this example binds bound names and
+ so is not respectful *)
+(*
+nominal_datatype trm =
+ Vr "name"
+| Lm x::"name" t::"trm" bind x in t
+| Lt left::"trm" right::"trm" bind "bv left" in right
+binder
+ bv
+where
+ "bv (Vr n) = {}"
+| "bv (Lm n t) = {atom n} \<union> bv t"
+| "bv (Lt l r) = bv l \<union> bv r"
+*)
+
+(* this example uses "-" in the binding function;
+ at the moment this is unsupported *)
+(*
+nominal_datatype trm' =
+ Vr' "name"
+| Lm' l::"name" r::"trm'" bind l in r
+| Lt' l::"trm'" r::"trm'" bind "bv' l" in r
+binder
+ bv'
+where
+ "bv' (Vr' n) = {atom n}"
+| "bv' (Lm' n t) = bv' t - {atom n}"
+| "bv' (Lt' l r) = bv' l \<union> bv' r"
+*)
+
+(* this example again binds bound names *)
+(*
+nominal_datatype trm'' =
+ Va'' "name"
+| Lm'' n::"name" l::"trm''" bind n in l
+and bla'' =
+ Bla'' f::"trm''" s::"trm''" bind "bv'' f" in s
+binder
+ bv''
+where
+ "bv'' (Vm'' x) = {}"
+| "bv'' (Lm'' x b) = {atom x}"
+*)
+
+end
+
+
+
--- a/Nominal/ROOT.ML Sun May 09 12:26:10 2010 +0100
+++ b/Nominal/ROOT.ML Sun May 09 12:38:59 2010 +0100
@@ -2,7 +2,7 @@
no_document use_thys
["Ex/Lambda",
- "Ex/ExLF",
+ "Ex/LF",
"Ex/SingleLet",
"Ex/Ex1rec",
"Ex/Ex2",
@@ -10,10 +10,10 @@
"Ex/ExLet",
"Ex/ExLetRec",
"Ex/TypeSchemes",
- "Ex/ExLeroy",
+ "Ex/Modules",
"Ex/ExPS3",
"Ex/ExPS7",
- "Ex/ExCoreHaskell",
+ "Ex/CoreHaskell",
"Ex/Test",
"Manual/Term4"
];