
diff -r 2f1b00d83925 -r b679900fa5f6 Nominal/Term5n.thy
--- a/Nominal/Term5n.thy	Tue Mar 23 08:16:39 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,228 +0,0 @@
-theory Term5
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"
-begin
-
-atom_decl name
-
-datatype rtrm5 =
-  rVr5 "name"
-| rAp5 "rtrm5" "rtrm5"
-| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"
-and rlts =
-  rLnil
-| rLcons "name" "rtrm5" "rlts"
-
-primrec
-  rbv5
-where
-  "rbv5 rLnil = {}"
-| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"
-
-
-setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Term5.rtrm5") 2 *}
-print_theorems
-
-local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Term5.rtrm5")
-  [[[], [], [(SOME (@{term rbv5}, false), 0, 1)]], [[], []]] [(@{term rbv5}, 1, [[], [0, 2]])] *}
-print_theorems
-
-notation
-  alpha_rtrm5 ("_ \<approx>5 _" [100, 100] 100) and
-  alpha_rlts ("_ \<approx>l _" [100, 100] 100)
-thm alpha_rtrm5_alpha_rlts_alpha_rbv5.intros
-
-local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_inj}, []), (build_alpha_inj @{thms alpha_rtrm5_alpha_rlts_alpha_rbv5.intros} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} ctxt)) ctxt)) *}
-thm alpha5_inj
-
-lemma rbv5_eqvt[eqvt]:
-  "pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
-  apply (induct x)
-  apply (simp_all add: eqvts atom_eqvt)
-  done
-
-lemma fv_rtrm5_rlts_eqvt[eqvt]:
-  "pi \<bullet> (fv_rtrm5 x) = fv_rtrm5 (pi \<bullet> x)"
-  "pi \<bullet> (fv_rlts l) = fv_rlts (pi \<bullet> l)"
-  "pi \<bullet> (fv_rbv5 l) = fv_rbv5 (pi \<bullet> l)"
-  apply (induct x and l)
-  apply (simp_all add: eqvts atom_eqvt)
-  done
-
-local_setup {*
-(fn ctxt => snd (Local_Theory.note ((@{binding alpha5_eqvt}, []),
-build_alpha_eqvts [@{term alpha_rtrm5}, @{term alpha_rlts}, @{term alpha_rbv5}] (fn _ => alpha_eqvt_tac  @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj permute_rtrm5_permute_rlts.simps} ctxt 1) ctxt) ctxt)) *}
-print_theorems
-
-lemma alpha5_reflp:
-"y \<approx>5 y \<and> (x \<approx>l x \<and> alpha_rbv5 x x)"
-apply (rule rtrm5_rlts.induct)
-apply (simp_all add: alpha5_inj)
-apply (rule_tac x="0::perm" in exI)
-apply (simp add: eqvts alpha_gen fresh_star_def fresh_zero_perm)
-done
-
-lemma alpha5_symp:
-"(a \<approx>5 b \<longrightarrow> b \<approx>5 a) \<and>
-(x \<approx>l y \<longrightarrow> y \<approx>l x) \<and>
-(alpha_rbv5 x y \<longrightarrow> alpha_rbv5 y x)"
-sorry
-
-lemma alpha5_transp:
-"(a \<approx>5 b \<longrightarrow> (\<forall>c. b \<approx>5 c \<longrightarrow> a \<approx>5 c)) \<and>
-(x \<approx>l y \<longrightarrow> (\<forall>z. y \<approx>l z \<longrightarrow> x \<approx>l z)) \<and>
-(alpha_rbv5 k l \<longrightarrow> (\<forall>m. alpha_rbv5 l m \<longrightarrow> alpha_rbv5 k m))"
-sorry
-
-lemma alpha5_equivp:
-  "equivp alpha_rtrm5"
-  "equivp alpha_rlts"
-  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
-  apply (simp_all only: alpha5_reflp)
-  apply (meson alpha5_symp alpha5_transp)
-  apply (meson alpha5_symp alpha5_transp)
-  done
-
-quotient_type
-  trm5 = rtrm5 / alpha_rtrm5
-and
-  lts = rlts / alpha_rlts
-  by (auto intro: alpha5_equivp)
-
-local_setup {*
-(fn ctxt => ctxt
- |> snd o (Quotient_Def.quotient_lift_const ("Vr5", @{term rVr5}))
- |> snd o (Quotient_Def.quotient_lift_const ("Ap5", @{term rAp5}))
- |> snd o (Quotient_Def.quotient_lift_const ("Lt5", @{term rLt5}))
- |> snd o (Quotient_Def.quotient_lift_const ("Lnil", @{term rLnil}))
- |> snd o (Quotient_Def.quotient_lift_const ("Lcons", @{term rLcons}))
- |> snd o (Quotient_Def.quotient_lift_const ("fv_trm5", @{term fv_rtrm5}))
- |> snd o (Quotient_Def.quotient_lift_const ("fv_lts", @{term fv_rlts}))
- |> snd o (Quotient_Def.quotient_lift_const ("fv_bv5", @{term fv_rbv5}))
- |> snd o (Quotient_Def.quotient_lift_const ("bv5", @{term rbv5}))
- |> snd o (Quotient_Def.quotient_lift_const ("alpha_bv5", @{term alpha_rbv5})))
-*}
-print_theorems
-
-lemma alpha5_rfv:
-  "(t \<approx>5 s \<Longrightarrow> fv_rtrm5 t = fv_rtrm5 s)"
-  "(l \<approx>l m \<Longrightarrow> (fv_rlts l = fv_rlts m \<and> fv_rbv5 l = fv_rbv5 m))"
-  "(alpha_rbv5 b c \<Longrightarrow> fv_rbv5 b = fv_rbv5 c)"
-  apply(induct rule: alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts)
-  apply(simp_all)
-  apply(simp add: alpha_gen)
-  done
-
-lemma bv_list_rsp:
-  shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"
-  apply(induct rule: alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts(2))
-  apply(simp_all)
-  apply(clarify)
-  apply simp
-  done
-
-local_setup {* snd o Local_Theory.note ((@{binding alpha_dis}, []), (flat (map (distinct_rel @{context} @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases}) [(@{thms rtrm5.distinct}, @{term alpha_rtrm5}), (@{thms rlts.distinct}, @{term alpha_rlts}), (@{thms rlts.distinct}, @{term alpha_rbv5})]))) *}
-print_theorems
-
-local_setup {* snd o Local_Theory.note ((@{binding alpha_bn_rsp}, []), prove_alpha_bn_rsp [@{term alpha_rtrm5}, @{term alpha_rlts}] @{thms alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts} @{thms alpha5_inj alpha_dis} @{thms alpha5_equivp} @{context} (@{term alpha_rbv5}, 1)) *}
-thm alpha_bn_rsp
-
-
-lemma [quot_respect]:
-  "(alpha_rlts ===> op =) fv_rlts fv_rlts"
-  "(alpha_rlts ===> op =) fv_rbv5 fv_rbv5"
-  "(alpha_rtrm5 ===> op =) fv_rtrm5 fv_rtrm5"
-  "(alpha_rlts ===> op =) rbv5 rbv5"
-  "(op = ===> alpha_rtrm5) rVr5 rVr5"
-  "(alpha_rtrm5 ===> alpha_rtrm5 ===> alpha_rtrm5) rAp5 rAp5"
-  "(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
-  "(op = ===> alpha_rtrm5 ===> alpha_rlts ===> alpha_rlts) rLcons rLcons"
-  "(op = ===> alpha_rtrm5 ===> alpha_rtrm5) permute permute"
-  "(op = ===> alpha_rlts ===> alpha_rlts) permute permute"
-  "(alpha_rlts ===> alpha_rlts ===> op =) alpha_rbv5 alpha_rbv5"
-  apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp alpha5_reflp alpha_bn_rsp)
-  apply (clarify)
-  apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
-done
-
-lemma
-  shows "(alpha_rlts ===> op =) rbv5 rbv5"
-  by (simp add: bv_list_rsp)
-
-lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]
-
-instantiation trm5 and lts :: pt
-begin
-
-quotient_definition
-  "permute_trm5 :: perm \<Rightarrow> trm5 \<Rightarrow> trm5"
-is
-  "permute :: perm \<Rightarrow> rtrm5 \<Rightarrow> rtrm5"
-
-quotient_definition
-  "permute_lts :: perm \<Rightarrow> lts \<Rightarrow> lts"
-is
-  "permute :: perm \<Rightarrow> rlts \<Rightarrow> rlts"
-
-instance by default
-  (simp_all add: permute_rtrm5_permute_rlts_zero[quot_lifted] permute_rtrm5_permute_rlts_append[quot_lifted])
-
-end
-
-lemmas permute_trm5_lts = permute_rtrm5_permute_rlts.simps[quot_lifted]
-lemmas bv5[simp] = rbv5.simps[quot_lifted]
-lemmas fv_trm5_bv5[simp] = fv_rtrm5_fv_rbv5.simps[quot_lifted]
-lemmas fv_lts[simp] = fv_rlts.simps[quot_lifted]
-lemmas alpha5_INJ = alpha5_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
-
-lemma lets_bla:
-  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Lt5 (Lcons x (Vr5 y) Lnil) (Vr5 x)) \<noteq> (Lt5 (Lcons x (Vr5 z) Lnil) (Vr5 x))"
-apply (simp only: alpha5_INJ)
-apply (simp only: bv5)
-apply simp
-done
-
-lemma lets_ok:
-  "(Lt5 (Lcons x (Vr5 y) Lnil) (Vr5 x)) = (Lt5 (Lcons y (Vr5 y) Lnil) (Vr5 y))"
-apply (simp add: alpha5_INJ)
-apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
-apply (simp_all add: alpha_gen)
-apply (simp add: permute_trm5_lts fresh_star_def eqvts)
-done
-
-lemma lets_ok3:
-  "x \<noteq> y \<Longrightarrow>
-   (Lt5 (Lcons x (Ap5 (Vr5 y) (Vr5 x)) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
-   (Lt5 (Lcons y (Ap5 (Vr5 x) (Vr5 y)) (Lcons x (Vr5 x) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
-apply (simp add: permute_trm5_lts alpha_gen alpha5_INJ)
-done
-
-
-lemma lets_not_ok1:
-  "(Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) =
-   (Lt5 (Lcons y (Vr5 x) (Lcons x (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
-apply (simp add: alpha5_INJ alpha_gen)
-apply (rule_tac x="0::perm" in exI)
-apply (simp add: permute_trm5_lts fresh_star_def alpha5_INJ(5) alpha5_INJ(2) alpha5_INJ(1) eqvts)
-apply blast
-done
-
-lemma distinct_helper:
-  shows "\<not>(rVr5 x \<approx>5 rAp5 y z)"
-  apply auto
-  apply (erule alpha_rtrm5.cases)
-  apply (simp_all only: rtrm5.distinct)
-  done
-
-lemma distinct_helper2:
-  shows "(Vr5 x) \<noteq> (Ap5 y z)"
-  by (lifting distinct_helper)
-
-lemma lets_nok:
-  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
-   (Lt5 (Lcons x (Ap5 (Vr5 z) (Vr5 z)) (Lcons y (Vr5 z) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
-   (Lt5 (Lcons y (Vr5 z) (Lcons x (Ap5 (Vr5 z) (Vr5 z)) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
-apply (simp only: alpha5_INJ(3) alpha5_INJ(5) alpha_gen permute_trm5_lts fresh_star_def)
-apply (simp add: distinct_helper2 alpha5_INJ permute_trm5_lts)
-done
-
-end




