--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Term1.thy Thu Feb 25 14:14:08 2010 +0100
@@ -0,0 +1,218 @@
+theory Term1
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"
+begin
+
+atom_decl name
+
+section {*** lets with binding patterns ***}
+
+datatype rtrm1 =
+ rVr1 "name"
+| rAp1 "rtrm1" "rtrm1"
+| rLm1 "name" "rtrm1" --"name is bound in trm1"
+| rLt1 "bp" "rtrm1" "rtrm1" --"all variables in bp are bound in the 2nd trm1"
+and bp =
+ BUnit
+| BVr "name"
+| BPr "bp" "bp"
+
+print_theorems
+
+(* to be given by the user *)
+
+primrec
+ bv1
+where
+ "bv1 (BUnit) = {}"
+| "bv1 (BVr x) = {atom x}"
+| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp2)"
+
+setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Term1.rtrm1", "Term1.bp"] *}
+thm permute_rtrm1_permute_bp.simps
+
+local_setup {*
+ snd o define_fv_alpha "Term1.rtrm1"
+ [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],
+ [[], [[]], [[], []]]] *}
+
+notation
+ alpha_rtrm1 ("_ \<approx>1 _" [100, 100] 100) and
+ alpha_bp ("_ \<approx>1b _" [100, 100] 100)
+thm alpha_rtrm1_alpha_bp.intros
+thm fv_rtrm1_fv_bp.simps
+
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha1_inj}, []), (build_alpha_inj @{thms alpha_rtrm1_alpha_bp.intros} @{thms rtrm1.distinct rtrm1.inject bp.distinct bp.inject} @{thms alpha_rtrm1.cases alpha_bp.cases} ctxt)) ctxt)) *}
+thm alpha1_inj
+
+local_setup {*
+snd o (build_eqvts @{binding bv1_eqvt} [@{term bv1}] [@{term "permute :: perm \<Rightarrow> bp \<Rightarrow> bp"}] (@{thms bv1.simps permute_rtrm1_permute_bp.simps}) @{thm rtrm1_bp.inducts(2)})
+*}
+
+local_setup {*
+snd o build_eqvts @{binding fv_rtrm1_fv_bp_eqvt} [@{term fv_rtrm1}, @{term fv_bp}] [@{term "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"},@{term "permute :: perm \<Rightarrow> bp \<Rightarrow> bp"}] (@{thms fv_rtrm1_fv_bp.simps permute_rtrm1_permute_bp.simps}) @{thm rtrm1_bp.induct}
+*}
+
+
+local_setup {*
+(fn ctxt => snd (Local_Theory.note ((@{binding alpha1_eqvt}, []),
+ build_alpha_eqvts [@{term alpha_rtrm1}, @{term alpha_bp}] [@{term "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"},@{term "permute :: perm \<Rightarrow> bp \<Rightarrow> bp"}] @{thms permute_rtrm1_permute_bp.simps alpha1_inj} @{thm alpha_rtrm1_alpha_bp.induct} ctxt) ctxt))
+*}
+print_theorems
+
+lemma alpha1_eqvt_proper[eqvt]:
+ "pi \<bullet> (t \<approx>1 s) = ((pi \<bullet> t) \<approx>1 (pi \<bullet> s))"
+ "pi \<bullet> (alpha_bp a b) = (alpha_bp (pi \<bullet> a) (pi \<bullet> b))"
+ apply (simp_all only: eqvts)
+ apply rule
+ apply (simp_all add: alpha1_eqvt)
+ apply (subst permute_minus_cancel(2)[symmetric,of "t" "pi"])
+ apply (subst permute_minus_cancel(2)[symmetric,of "s" "pi"])
+ apply (simp_all only: alpha1_eqvt)
+ apply rule
+ apply (simp_all add: alpha1_eqvt)
+ apply (subst permute_minus_cancel(2)[symmetric,of "a" "pi"])
+ apply (subst permute_minus_cancel(2)[symmetric,of "b" "pi"])
+ apply (simp_all only: alpha1_eqvt)
+done
+
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha1_equivp}, []),
+ (build_equivps [@{term alpha_rtrm1}, @{term alpha_bp}] @{thm rtrm1_bp.induct} @{thm alpha_rtrm1_alpha_bp.induct} @{thms rtrm1.inject bp.inject} @{thms alpha1_inj} @{thms rtrm1.distinct bp.distinct} @{thms alpha_rtrm1.cases alpha_bp.cases} @{thms alpha1_eqvt} ctxt)) ctxt)) *}
+thm alpha1_equivp
+
+local_setup {* define_quotient_type [(([], @{binding trm1}, NoSyn), (@{typ rtrm1}, @{term alpha_rtrm1}))]
+ (rtac @{thm alpha1_equivp(1)} 1) *}
+
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr1", @{term rVr1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Ap1", @{term rAp1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lm1", @{term rLm1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lt1", @{term rLt1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm1", @{term fv_rtrm1})))
+*}
+print_theorems
+
+thm alpha_rtrm1_alpha_bp.induct
+local_setup {* prove_const_rsp @{binding fv_rtrm1_rsp} [@{term fv_rtrm1}]
+ (fn _ => fvbv_rsp_tac @{thm alpha_rtrm1_alpha_bp.inducts(1)} @{thms fv_rtrm1_fv_bp.simps} 1) *}
+local_setup {* prove_const_rsp @{binding rVr1_rsp} [@{term rVr1}]
+ (fn _ => constr_rsp_tac @{thms alpha1_inj} @{thms fv_rtrm1_rsp} @{thms alpha1_equivp} 1) *}
+local_setup {* prove_const_rsp @{binding rAp1_rsp} [@{term rAp1}]
+ (fn _ => constr_rsp_tac @{thms alpha1_inj} @{thms fv_rtrm1_rsp} @{thms alpha1_equivp} 1) *}
+local_setup {* prove_const_rsp @{binding rLm1_rsp} [@{term rLm1}]
+ (fn _ => constr_rsp_tac @{thms alpha1_inj} @{thms fv_rtrm1_rsp} @{thms alpha1_equivp} 1) *}
+local_setup {* prove_const_rsp @{binding rLt1_rsp} [@{term rLt1}]
+ (fn _ => constr_rsp_tac @{thms alpha1_inj} @{thms fv_rtrm1_rsp} @{thms alpha1_equivp} 1) *}
+local_setup {* prove_const_rsp @{binding permute_rtrm1_rsp} [@{term "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"}]
+ (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha1_eqvt}) 1) *}
+
+lemmas trm1_bp_induct = rtrm1_bp.induct[quot_lifted]
+lemmas trm1_bp_inducts = rtrm1_bp.inducts[quot_lifted]
+
+setup {* define_lifted_perms ["Term1.trm1"] [("permute_trm1", @{term "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"})]
+ @{thms permute_rtrm1_permute_bp_zero permute_rtrm1_permute_bp_append} *}
+
+lemmas
+ permute_trm1 = permute_rtrm1_permute_bp.simps[quot_lifted]
+and fv_trm1 = fv_rtrm1_fv_bp.simps[quot_lifted]
+and fv_trm1_eqvt = fv_rtrm1_fv_bp_eqvt(1)[quot_lifted]
+and alpha1_INJ = alpha1_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+
+lemma supports:
+ "(supp (atom x)) supports (Vr1 x)"
+ "(supp t \<union> supp s) supports (Ap1 t s)"
+ "(supp (atom x) \<union> supp t) supports (Lm1 x t)"
+ "(supp b \<union> supp t \<union> supp s) supports (Lt1 b t s)"
+ "{} supports BUnit"
+ "(supp (atom x)) supports (BVr x)"
+ "(supp a \<union> supp b) supports (BPr a b)"
+apply(simp_all add: supports_def fresh_def[symmetric] swap_fresh_fresh permute_trm1)
+apply(rule_tac [!] allI)+
+apply(rule_tac [!] impI)
+apply(tactic {* ALLGOALS (REPEAT o etac conjE) *})
+apply(simp_all add: fresh_atom)
+done
+
+lemma rtrm1_bp_fs:
+ "finite (supp (x :: trm1))"
+ "finite (supp (b :: bp))"
+ apply (induct x and b rule: trm1_bp_inducts)
+ apply(tactic {* ALLGOALS (rtac @{thm supports_finite} THEN' resolve_tac @{thms supports}) *})
+ apply(simp_all add: supp_atom)
+ done
+
+instance trm1 :: fs
+apply default
+apply (rule rtrm1_bp_fs(1))
+done
+
+lemma fv_eq_bv: "fv_bp bp = bv1 bp"
+apply(induct bp rule: trm1_bp_inducts(2))
+apply(simp_all)
+done
+
+lemma helper2: "{b. \<forall>pi. pi \<bullet> (a \<rightleftharpoons> b) \<bullet> bp \<noteq> bp} = {}"
+apply auto
+apply (rule_tac x="(x \<rightleftharpoons> a)" in exI)
+apply auto
+done
+
+lemma alpha_bp_eq_eq: "alpha_bp a b = (a = b)"
+apply rule
+apply (induct a b rule: alpha_rtrm1_alpha_bp.inducts(2))
+apply (simp_all add: equivp_reflp[OF alpha1_equivp(2)])
+done
+
+lemma alpha_bp_eq: "alpha_bp = (op =)"
+apply (rule ext)+
+apply (rule alpha_bp_eq_eq)
+done
+
+lemma supp_fv:
+ "supp t = fv_trm1 t"
+ "supp b = fv_bp b"
+apply(induct t and b rule: trm1_bp_inducts)
+apply(simp_all)
+apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
+apply(simp only: supp_at_base[simplified supp_def])
+apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
+apply(simp add: Collect_imp_eq Collect_neg_eq)
+apply(subgoal_tac "supp (Lm1 name rtrm1) = supp (Abs {atom name} rtrm1)")
+apply(simp add: supp_Abs fv_trm1)
+apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt permute_trm1)
+apply(simp add: alpha1_INJ)
+apply(simp add: Abs_eq_iff)
+apply(simp add: alpha_gen.simps)
+apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric])
+apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp(rtrm11) \<union> supp (Abs (bv1 bp) rtrm12)")
+apply(simp add: supp_Abs fv_trm1 fv_eq_bv)
+apply(simp (no_asm) add: supp_def permute_trm1)
+apply(simp add: alpha1_INJ alpha_bp_eq)
+apply(simp add: Abs_eq_iff)
+apply(simp add: alpha_gen)
+apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
+apply(simp add: Collect_imp_eq Collect_neg_eq fresh_star_def helper2)
+apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
+apply(simp (no_asm) add: supp_def eqvts)
+apply(fold supp_def)
+apply(simp add: supp_at_base)
+apply(simp (no_asm) add: supp_def Collect_imp_eq Collect_neg_eq)
+apply(simp add: Collect_imp_eq[symmetric] Collect_neg_eq[symmetric] supp_def[symmetric])
+done
+
+lemma trm1_supp:
+ "supp (Vr1 x) = {atom x}"
+ "supp (Ap1 t1 t2) = supp t1 \<union> supp t2"
+ "supp (Lm1 x t) = (supp t) - {atom x}"
+ "supp (Lt1 b t s) = supp t \<union> (supp s - bv1 b)"
+by (simp_all add: supp_fv fv_trm1 fv_eq_bv)
+
+lemma trm1_induct_strong:
+ assumes "\<And>name b. P b (Vr1 name)"
+ and "\<And>rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12\<rbrakk> \<Longrightarrow> P b (Ap1 rtrm11 rtrm12)"
+ and "\<And>name rtrm1 b. \<lbrakk>\<And>c. P c rtrm1; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lm1 name rtrm1)"
+ and "\<And>bp rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12; bv1 bp \<sharp>* b\<rbrakk> \<Longrightarrow> P b (Lt1 bp rtrm11 rtrm12)"
+ shows "P a rtrma"
+sorry
+
+end