--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Quot/Examples/AbsRepTest.thy Thu Feb 25 07:57:17 2010 +0100
@@ -0,0 +1,240 @@
+theory AbsRepTest
+imports "../Quotient" "../Quotient_List" "../Quotient_Option" "../Quotient_Sum" "../Quotient_Product" List
+begin
+
+
+(*
+ML_command "ProofContext.debug := false"
+ML_command "ProofContext.verbose := false"
+*)
+
+ML {* open Quotient_Term *}
+
+ML {*
+fun test_funs flag ctxt (rty, qty) =
+ (absrep_fun_chk flag ctxt (rty, qty)
+ |> Syntax.string_of_term ctxt
+ |> writeln;
+ equiv_relation_chk ctxt (rty, qty)
+ |> Syntax.string_of_term ctxt
+ |> writeln)
+*}
+
+definition
+ erel1 (infixl "\<approx>1" 50)
+where
+ "erel1 \<equiv> \<lambda>xs ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"
+
+quotient_type
+ 'a fset = "'a list" / erel1
+ apply(rule equivpI)
+ unfolding erel1_def reflp_def symp_def transp_def
+ by auto
+
+definition
+ erel2 (infixl "\<approx>2" 50)
+where
+ "erel2 \<equiv> \<lambda>(xs::('a * 'a) list) ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"
+
+quotient_type
+ 'a foo = "('a * 'a) list" / erel2
+ apply(rule equivpI)
+ unfolding erel2_def reflp_def symp_def transp_def
+ by auto
+
+definition
+ erel3 (infixl "\<approx>3" 50)
+where
+ "erel3 \<equiv> \<lambda>(xs::('a * int) list) ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"
+
+quotient_type
+ 'a bar = "('a * int) list" / "erel3"
+ apply(rule equivpI)
+ unfolding erel3_def reflp_def symp_def transp_def
+ by auto
+
+fun
+ intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infixl "\<approx>4" 50)
+where
+ "intrel (x, y) (u, v) = (x + v = u + y)"
+
+quotient_type myint = "nat \<times> nat" / intrel
+ by (auto simp add: equivp_def expand_fun_eq)
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "nat \<times> nat"},
+ @{typ "myint"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "('a * 'a) list"},
+ @{typ "'a foo"})
+*}
+
+ML {*
+test_funs RepF @{context}
+ (@{typ "(('a * 'a) list * 'b)"},
+ @{typ "('a foo * 'b)"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "(('a list) * int) list"},
+ @{typ "('a fset) bar"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "('a list)"},
+ @{typ "('a fset)"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "('a list) list"},
+ @{typ "('a fset) fset"})
+*}
+
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "((nat * nat) list) list"},
+ @{typ "((myint) fset) fset"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "(('a * 'a) list) list"},
+ @{typ "(('a * 'a) fset) fset"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "(nat * nat) list"},
+ @{typ "myint fset"})
+*}
+
+ML {*
+test_funs AbsF @{context}
+ (@{typ "('a list) list \<Rightarrow> 'a list"},
+ @{typ "('a fset) fset \<Rightarrow> 'a fset"})
+*}
+
+lemma OO_sym_inv:
+ assumes sr: "symp r"
+ and ss: "symp s"
+ shows "(r OO s) x y = (s OO r) y x"
+ using sr ss
+ unfolding symp_def
+ apply (metis pred_comp.intros pred_compE ss symp_def)
+ done
+
+lemma abs_o_rep:
+ assumes a: "Quotient r absf repf"
+ shows "absf o repf = id"
+ apply(rule ext)
+ apply(simp add: Quotient_abs_rep[OF a])
+ done
+
+lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"
+ apply (rule eq_reflection)
+ apply auto
+ done
+
+lemma map_rel_cong: "b \<approx>1 ba \<Longrightarrow> map f b \<approx>1 map f ba"
+ unfolding erel1_def
+ apply(simp only: set_map set_in_eq)
+ done
+
+lemma quotient_compose_list_gen_pre:
+ assumes a: "equivp r2"
+ and b: "Quotient r2 abs2 rep2"
+ shows "(list_rel r2 OOO op \<approx>1) r s =
+ ((list_rel r2 OOO op \<approx>1) r r \<and> (list_rel r2 OOO op \<approx>1) s s \<and>
+ abs_fset (map abs2 r) = abs_fset (map abs2 s))"
+ apply rule
+ apply rule
+ apply rule
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply rule
+ apply (rule equivp_reflp[OF fset_equivp])
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply(rule)
+ apply rule
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply rule
+ apply (rule equivp_reflp[OF fset_equivp])
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply (subgoal_tac "map abs2 r \<approx>1 map abs2 s")
+ apply (metis Quotient_rel[OF Quotient_fset])
+ apply (auto)[1]
+ apply (subgoal_tac "map abs2 r = map abs2 b")
+ prefer 2
+ apply (metis Quotient_rel[OF list_quotient[OF b]])
+ apply (subgoal_tac "map abs2 s = map abs2 ba")
+ prefer 2
+ apply (metis Quotient_rel[OF list_quotient[OF b]])
+ apply (simp add: map_rel_cong)
+ apply rule
+ apply (rule rep_abs_rsp[of "list_rel r2" "map abs2"])
+ apply (rule list_quotient)
+ apply (rule b)
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply rule
+ prefer 2
+ apply (rule rep_abs_rsp_left[of "list_rel r2" "map abs2"])
+ apply (rule list_quotient)
+ apply (rule b)
+ apply (rule list_rel_refl)
+ apply (metis equivp_def a)
+ apply (erule conjE)+
+ apply (subgoal_tac "map abs2 r \<approx>1 map abs2 s")
+ apply (rule map_rel_cong)
+ apply (assumption)
+ apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp a b)
+ done
+
+lemma quotient_compose_list_gen:
+ assumes a: "Quotient r2 abs2 rep2"
+ and b: "equivp r2" (* reflp is not enough *)
+ shows "Quotient ((list_rel r2) OOO (op \<approx>1))
+ (abs_fset \<circ> (map abs2)) ((map rep2) \<circ> rep_fset)"
+ unfolding Quotient_def comp_def
+ apply (rule)+
+ apply (simp add: abs_o_rep[OF a] id_simps Quotient_abs_rep[OF Quotient_fset])
+ apply (rule)
+ apply (rule)
+ apply (rule)
+ apply (rule list_rel_refl)
+ apply (metis b equivp_def)
+ apply (rule)
+ apply (rule equivp_reflp[OF fset_equivp])
+ apply (rule list_rel_refl)
+ apply (metis b equivp_def)
+ apply rule
+ apply rule
+ apply(rule quotient_compose_list_gen_pre[OF b a])
+ done
+
+(* This is the general statement but the types of abs2 and rep2
+ are wrong as can be seen in following exanples *)
+lemma quotient_compose_general:
+ assumes a2: "Quotient r1 abs1 rep1"
+ and "Quotient r2 abs2 rep2"
+ shows "Quotient ((list_rel r2) OOO r1)
+ (abs1 \<circ> (map abs2)) ((map rep2) \<circ> rep1)"
+sorry
+
+thm quotient_compose_list_gen[OF Quotient_fset fset_equivp]
+thm quotient_compose_general[OF Quotient_fset]
+(* Doesn't work: *)
+(* thm quotient_compose_general[OF Quotient_fset Quotient_fset] *)
+
+end