--- a/IntEx.thy	Sun Dec 06 00:13:35 2009 +0100
+++ b/IntEx.thy	Sun Dec 06 00:19:45 2009 +0100
@@ -162,7 +162,6 @@
 apply(tactic {* procedure_tac @{context} @{thm test1} 1 *})
 apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
 apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
-apply(tactic {* simp_tac (HOL_basic_ss addSolver quotient_solver addsimps @{thms all_prs})  1 *})
 apply(tactic {* clean_tac @{context} 1 *}) 
 done
 
@@ -217,22 +216,6 @@
 apply simp
 done
 
-(* I believe it's true. *)
-lemma foldl_rsp[quotient_rsp]:
-  "((op \<approx> ===> op \<approx> ===> op \<approx>) ===> op \<approx> ===> list_rel op \<approx> ===> op \<approx>) foldl foldl"
-apply (simp only: fun_rel.simps)
-apply (rule allI)
-apply (rule allI)
-apply (rule impI)
-apply (rule allI)
-apply (rule allI)
-apply (rule impI)
-apply (rule allI)
-apply (rule allI)
-apply (rule impI)
-apply (induct_tac xb yb rule: list_induct2) (* To finish I need to give it: arbitrary:xa ya *)
-sorry
-
 lemma "foldl PLUS x [] = x"
 apply(tactic {* procedure_tac @{context} @{thm ho_tst} 1 *})
 apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})

--- a/LFex.thy	Sun Dec 06 00:13:35 2009 +0100
+++ b/LFex.thy	Sun Dec 06 00:19:45 2009 +0100
@@ -220,6 +220,7 @@
   val quot = @{thms Quotient_KIND Quotient_TY Quotient_TRM}
   val rel_refl = map (fn x => @{thm equivp_reflp} OF [x]) @{thms alpha_equivps}
   val reps_same = map (fn x => @{thm Quotient_rel_rep} OF [x]) quot
+  val trans2 = map (fn x => @{thm equals_rsp} OF [x]) quot
 *}
 
 lemma 
@@ -256,12 +257,11 @@
          ((x5 :: TRM) = x6 \<longrightarrow> P3 x5 x6)"
 using a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11
 apply - 
-apply(tactic {* lift_tac @{context} @{thm akind_aty_atrm.induct} @{thms alpha_equivps} 1 *})
-(*apply(tactic {* regularize_tac @{context} @{thms alpha_equivps} 1 *})
+apply(tactic {* procedure_tac @{context} @{thm akind_aty_atrm.induct} 1 *})
+apply(tactic {* regularize_tac @{context} @{thms alpha_equivps} 1 *})
 apply(tactic {* all_inj_repabs_tac @{context} rel_refl trans2 1 *})
+apply(fold perm_kind_def perm_ty_def perm_trm_def FV_ty_def[simplified id_simps] FV_kind_def[simplified id_simps] FV_trm_def[simplified id_simps])
 apply(tactic {* clean_tac @{context} 1 *})
-*)
-(*apply(tactic {* all_inj_repabs_tac @{context} rel_refl trans2 1 *})*)
 (*
 Profiling:
 ML_prf {* fun ith i =  (#concl (fst (Subgoal.focus @{context} i (#goal (Isar.goal ()))))) *}
@@ -270,8 +270,6 @@
 ML_prf {* PolyML.profiling 1 *}
 ML_prf {* profile 2 Seq.list_of ((all_inj_repabs_tac @{context} quot rel_refl trans2 1) (#goal (Isar.goal ()))) *}
 *)
-apply(unfold perm_kind_def perm_ty_def perm_trm_def)
-apply(tactic {* clean_tac @{context} 1 *})
 done
 
 (* Does not work:

--- a/LamEx.thy	Sun Dec 06 00:13:35 2009 +0100
+++ b/LamEx.thy	Sun Dec 06 00:19:45 2009 +0100
@@ -190,14 +190,20 @@
 
 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"
@@ -210,6 +216,8 @@
 
 lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
 apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
@@ -217,6 +225,8 @@
      \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
     \<Longrightarrow> P"
 apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);
@@ -225,6 +235,8 @@
         \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
     \<Longrightarrow> qxb qx qxa"
 apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma var_inject: "(Var a = Var b) = (a = b)"

--- a/QuotList.thy	Sun Dec 06 00:13:35 2009 +0100
+++ b/QuotList.thy	Sun Dec 06 00:19:45 2009 +0100
@@ -84,10 +84,22 @@
   shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
 by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
 
-(* We need an ho version *)
 lemma map_rsp:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
+  shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"
+apply(simp)
+apply(rule allI)+
+apply(rule impI)
+apply(rule allI)+
+apply (induct_tac xa ya rule: list_induct2')
+apply simp_all
+done
+
+(* TODO: if the above is correct, we can remove this one *)
+lemma map_rsp_lo:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   and     a: "(R1 ===> R2) f1 f2"
   and     b: "list_rel R1 l1 l2"
   shows "list_rel R2 (map f1 l1) (map f2 l2)"
@@ -106,8 +118,35 @@
   shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
 by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
 
+lemma list_rel_empty: "list_rel R [] b \<Longrightarrow> length b = 0"
+by (induct b) (simp_all)
 
+lemma list_rel_len: "list_rel R a b \<Longrightarrow> length a = length b"
+apply (induct a arbitrary: b)
+apply (simp add: list_rel_empty)
+apply (case_tac b)
+apply simp_all
+done
 
+(* TODO: induct_tac doesn't accept 'arbitrary'.
+         induct     doesn't accept 'rule'.
+   that's why the proof uses manual generalisation and needs assumptions
+   both in conclusion for induction and in assumptions. *)
+lemma foldl_rsp:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"
+apply auto
+apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
+apply simp
+apply (rule_tac x="xa" in spec)
+apply (rule_tac x="ya" in spec)
+apply (rule_tac xs="xb" and ys="yb" in list_induct2)
+apply (rule list_rel_len)
+apply (simp_all)
+done
+
+(* TODO: foldr_rsp should be similar *)
 
 
 

--- a/QuotMain.thy	Sun Dec 06 00:13:35 2009 +0100
+++ b/QuotMain.thy	Sun Dec 06 00:19:45 2009 +0100
@@ -149,7 +149,7 @@
   fun_quotient list_quotient
 
 lemmas [quotient_rsp] =
-  quot_rel_rsp nil_rsp cons_rsp
+  quot_rel_rsp nil_rsp cons_rsp foldl_rsp
 
 ML {* maps_lookup @{theory} "List.list" *}
 ML {* maps_lookup @{theory} "*" *}
@@ -1110,11 +1110,10 @@
 fun clean_tac lthy =
   let
     val thy = ProofContext.theory_of lthy;
-    val defs = map (Thm.varifyT o #def) (qconsts_dest thy)
-    val thms1 = @{thms all_prs ex_prs}
+    val defs = map (Thm.varifyT o symmetric o #def) (qconsts_dest thy)
+    val thms1 = @{thms all_prs ex_prs} @ defs
     val thms2 = @{thms eq_reflection[OF fun_map.simps]} 
                 @ @{thms id_simps Quotient_abs_rep Quotient_rel_rep} 
-                @ defs
     fun simp_ctxt thms = HOL_basic_ss addsimps thms addSolver quotient_solver
     (* FIXME: use of someting smaller than HOL_basic_ss *)
   in