
diff -r 20f0b148cfe2 -r 3da18bf6886c FSet.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/FSet.thy	Fri Oct 23 16:34:20 2009 +0200
@@ -0,0 +1,713 @@
+theory FSet
+imports QuotMain
+begin
+
+inductive
+  list_eq (infix "\<approx>" 50)
+where
+  "a#b#xs \<approx> b#a#xs"
+| "[] \<approx> []"
+| "xs \<approx> ys \<Longrightarrow> ys \<approx> xs"
+| "a#a#xs \<approx> a#xs"
+| "xs \<approx> ys \<Longrightarrow> a#xs \<approx> a#ys"
+| "\<lbrakk>xs1 \<approx> xs2; xs2 \<approx> xs3\<rbrakk> \<Longrightarrow> xs1 \<approx> xs3"
+
+lemma list_eq_refl:
+  shows "xs \<approx> xs"
+  apply (induct xs)
+   apply (auto intro: list_eq.intros)
+  done
+
+lemma equiv_list_eq:
+  shows "EQUIV list_eq"
+  unfolding EQUIV_REFL_SYM_TRANS REFL_def SYM_def TRANS_def
+  apply(auto intro: list_eq.intros list_eq_refl)
+  done
+
+quotient fset = "'a list" / "list_eq"
+  apply(rule equiv_list_eq)
+  done
+
+print_theorems
+
+typ "'a fset"
+thm "Rep_fset"
+thm "ABS_fset_def"
+
+ML {* @{term "Abs_fset"} *}
+local_setup {*
+  make_const_def @{binding EMPTY} @{term "[]"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+term Nil
+term EMPTY
+thm EMPTY_def
+
+
+local_setup {*
+  make_const_def @{binding INSERT} @{term "op #"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+term Cons
+term INSERT
+thm INSERT_def
+
+local_setup {*
+  make_const_def @{binding UNION} @{term "op @"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+term append
+term UNION
+thm UNION_def
+
+
+thm QUOTIENT_fset
+
+thm QUOT_TYPE_I_fset.thm11
+
+
+fun
+  membship :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infix "memb" 100)
+where
+  m1: "(x memb []) = False"
+| m2: "(x memb (y#xs)) = ((x=y) \<or> (x memb xs))"
+
+fun
+  card1 :: "'a list \<Rightarrow> nat"
+where
+  card1_nil: "(card1 []) = 0"
+| card1_cons: "(card1 (x # xs)) = (if (x memb xs) then (card1 xs) else (Suc (card1 xs)))"
+
+local_setup {*
+  make_const_def @{binding card} @{term "card1"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+term card1
+term card
+thm card_def
+
+(* text {*
+ Maybe make_const_def should require a theorem that says that the particular lifted function
+ respects the relation. With it such a definition would be impossible:
+ make_const_def @{binding CARD} @{term "length"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}*)
+
+lemma card1_0:
+  fixes a :: "'a list"
+  shows "(card1 a = 0) = (a = [])"
+  apply (induct a)
+   apply (simp)
+  apply (simp_all)
+   apply meson
+  apply (simp_all)
+  done
+
+lemma not_mem_card1:
+  fixes x :: "'a"
+  fixes xs :: "'a list"
+  shows "~(x memb xs) \<Longrightarrow> card1 (x # xs) = Suc (card1 xs)"
+  by simp
+
+
+lemma mem_cons:
+  fixes x :: "'a"
+  fixes xs :: "'a list"
+  assumes a : "x memb xs"
+  shows "x # xs \<approx> xs"
+  using a
+  apply (induct xs)
+  apply (auto intro: list_eq.intros)
+  done
+
+lemma card1_suc:
+  fixes xs :: "'a list"
+  fixes n :: "nat"
+  assumes c: "card1 xs = Suc n"
+  shows "\<exists>a ys. ~(a memb ys) \<and> xs \<approx> (a # ys)"
+  using c
+apply(induct xs)
+apply (metis Suc_neq_Zero card1_0)
+apply (metis QUOT_TYPE_I_fset.R_trans card1_cons list_eq_refl mem_cons)
+done
+
+primrec
+  fold1
+where
+  "fold1 f (g :: 'a \<Rightarrow> 'b) (z :: 'b) [] = z"
+| "fold1 f g z (a # A) =
+     (if ((!u v. (f u v = f v u))
+      \<and> (!u v w. ((f u (f v w) = f (f u v) w))))
+     then (
+       if (a memb A) then (fold1 f g z A) else (f (g a) (fold1 f g z A))
+     ) else z)"
+
+(* fold1_def is not usable, but: *)
+thm fold1.simps
+
+lemma fs1_strong_cases:
+  fixes X :: "'a list"
+  shows "(X = []) \<or> (\<exists>a. \<exists> Y. (~(a memb Y) \<and> (X \<approx> a # Y)))"
+  apply (induct X)
+  apply (simp)
+  apply (metis QUOT_TYPE_I_fset.thm11 list_eq_refl mem_cons m1)
+  done
+
+local_setup {*
+  make_const_def @{binding IN} @{term "membship"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+term membship
+term IN
+thm IN_def
+
+ML {*
+  val consts = [@{const_name "Nil"}, @{const_name "Cons"},
+                @{const_name "membship"}, @{const_name "card1"},
+                @{const_name "append"}, @{const_name "fold1"}];
+*}
+
+ML {* val fset_defs = @{thms EMPTY_def IN_def UNION_def card_def INSERT_def} *}
+ML {* val fset_defs_sym = map (fn t => symmetric (unabs_def @{context} t)) fset_defs *}
+
+thm fun_map.simps
+text {* Respectfullness *}
+
+lemma mem_respects:
+  fixes z
+  assumes a: "list_eq x y"
+  shows "(z memb x) = (z memb y)"
+  using a by induct auto
+
+lemma card1_rsp:
+  fixes a b :: "'a list"
+  assumes e: "a \<approx> b"
+  shows "card1 a = card1 b"
+  using e apply induct
+  apply (simp_all add:mem_respects)
+  done
+
+lemma cons_preserves:
+  fixes z
+  assumes a: "xs \<approx> ys"
+  shows "(z # xs) \<approx> (z # ys)"
+  using a by (rule list_eq.intros(5))
+
+lemma append_respects_fst:
+  assumes a : "list_eq l1 l2"
+  shows "list_eq (l1 @ s) (l2 @ s)"
+  using a
+  apply(induct)
+  apply(auto intro: list_eq.intros)
+  apply(simp add: list_eq_refl)
+done
+
+thm list.induct
+lemma list_induct_hol4:
+  fixes P :: "'a list \<Rightarrow> bool"
+  assumes "((P []) \<and> (\<forall>t. (P t) \<longrightarrow> (\<forall>h. (P (h # t)))))"
+  shows "(\<forall>l. (P l))"
+  sorry
+
+ML {* atomize_thm @{thm list_induct_hol4} *}
+
+prove list_induct_r: {*
+   build_regularize_goal (atomize_thm @{thm list_induct_hol4}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}
+  apply (simp only: equiv_res_forall[OF equiv_list_eq])
+  thm RIGHT_RES_FORALL_REGULAR
+  apply (rule RIGHT_RES_FORALL_REGULAR)
+  prefer 2
+  apply (assumption)
+  apply (metis)
+  done
+
+ML {*
+fun instantiate_tac thm = Subgoal.FOCUS (fn {concl, ...} =>
+let
+  val pat = Drule.strip_imp_concl (cprop_of thm)
+  val insts = Thm.match (pat, concl)
+in
+  rtac (Drule.instantiate insts thm) 1
+end
+handle _ => no_tac
+)
+*}
+
+ML {*
+fun res_forall_rsp_tac ctxt =
+  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))
+  THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}
+  THEN' instantiate_tac @{thm RES_FORALL_RSP} ctxt THEN'
+  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))
+*}
+
+
+ML {*
+fun quotient_tac quot_thm =
+  REPEAT_ALL_NEW (FIRST' [
+    rtac @{thm FUN_QUOTIENT},
+    rtac quot_thm,
+    rtac @{thm IDENTITY_QUOTIENT}
+  ])
+*}
+
+ML {*
+fun r_mk_comb_tac ctxt quot_thm reflex_thm trans_thm =
+  (FIRST' [
+    rtac @{thm FUN_QUOTIENT},
+    rtac quot_thm,
+    rtac @{thm IDENTITY_QUOTIENT},
+    rtac @{thm ext},
+    rtac trans_thm,
+    res_forall_rsp_tac ctxt,
+    (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),
+    (instantiate_tac @{thm APPLY_RSP} ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),
+    rtac reflex_thm,
+    atac,
+    (
+     (simp_tac ((Simplifier.context @{context} HOL_ss) addsimps @{thms FUN_REL.simps}))
+     THEN_ALL_NEW (fn _ => no_tac)
+    )
+    ])
+*}
+
+ML {*
+fun r_mk_comb_tac_fset ctxt =
+  r_mk_comb_tac ctxt @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}
+*}
+
+
+ML {* val thm = @{thm list_induct_r} OF [atomize_thm @{thm list_induct_hol4}] *}
+ML {* val trm_r = build_repabs_goal @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}
+ML {* val trm = build_repabs_term @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}
+
+prove list_induct_tr: trm_r
+apply (atomize(full))
+(* APPLY_RSP_TAC *)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+(* LAMBDA_RES_TAC *)
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* REFL_TAC *)
+apply (simp)
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* REFL_TAC *)
+apply (simp)
+(* APPLY_RSP_TAC *)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* REFL_TAC *)
+apply (simp)
+(* APPLY_RSP *)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+(* MK_COMB_TAC *)
+apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+(* REFL_TAC *)
+apply (simp)
+(* W(C (curry op THEN) (G... *)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+(* CONS respects *)
+apply (simp add: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+apply (rule cons_preserves)
+apply (assumption)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+done
+
+prove list_induct_t: trm
+apply (simp only: list_induct_tr[symmetric])
+apply (tactic {* rtac thm 1 *})
+done
+
+ML {* val nthm = MetaSimplifier.rewrite_rule fset_defs_sym (snd (no_vars (Context.Theory @{theory}, @{thm list_induct_t}))) *}
+
+thm list.recs(2)
+thm m2
+ML {* atomize_thm @{thm m2} *}
+
+prove m2_r_p: {*
+   build_regularize_goal (atomize_thm @{thm m2}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}
+  apply (simp add: equiv_res_forall[OF equiv_list_eq])
+done
+
+ML {* val m2_r = @{thm m2_r_p} OF [atomize_thm @{thm m2}] *}
+ML {* val m2_t_g = build_repabs_goal @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}
+ML {* val m2_t = build_repabs_term @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}
+prove m2_t_p: m2_t_g
+apply (atomize(full))
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+apply (simp add: FUN_REL.simps)
+prefer 2
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+prefer 2
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+apply (simp only: FUN_REL.simps)
+apply (rule allI)
+apply (rule allI)
+apply (rule impI)
+apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
+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 (simp add:mem_respects)
+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 (simp add:cons_preserves)
+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 (simp add:mem_respects)
+apply (auto)
+done
+
+prove m2_t: m2_t
+apply (simp only: m2_t_p[symmetric])
+apply (tactic {* rtac m2_r 1 *})
+done
+
+lemma id_apply2 [simp]: "id x \<equiv> x"
+  by (simp add: id_def)
+
+
+thm LAMBDA_PRS
+ML {*
+ val t = prop_of @{thm LAMBDA_PRS}
+ val tt = Drule.instantiate' [SOME @{ctyp "'a list"}, SOME @{ctyp "'a fset"}] [] @{thm LAMBDA_PRS}
+ val ttt = @{thm LAMBDA_PRS} OF [@{thm QUOTIENT_fset}, @{thm IDENTITY_QUOTIENT}]
+ val tttt = @{thm "HOL.sym"} OF [ttt]
+*}
+ML {*
+ val ttttt = MetaSimplifier.rewrite_rule [@{thm id_apply2}] tttt
+ val zzz = @{thm m2_t}
+*}
+
+ML {*
+fun eqsubst_thm ctxt thms thm =
+  let
+    val goalstate = Goal.init (Thm.cprop_of thm)
+    val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of
+      NONE => error "eqsubst_thm"
+    | SOME th => cprem_of th 1
+    val tac = (EqSubst.eqsubst_tac ctxt [0] thms 1) THEN simp_tac HOL_ss 1
+    val cgoal = cterm_of (ProofContext.theory_of ctxt) (Logic.mk_equals (term_of (Thm.cprop_of thm), term_of a'))
+    val rt = Toplevel.program (fn () => Goal.prove_internal [] cgoal (fn _ => tac));
+  in
+    Simplifier.rewrite_rule [rt] thm
+  end
+*}
+ML {* val m2_t' = eqsubst_thm @{context} [ttttt] @{thm m2_t} *}
+ML {* val rwr = @{thm FORALL_PRS[OF QUOTIENT_fset]} *}
+ML {* val rwrs = @{thm "HOL.sym"} OF [ObjectLogic.rulify rwr] *}
+ML {* rwrs; m2_t' *}
+ML {* eqsubst_thm @{context} [rwrs] m2_t' *}
+thm INSERT_def
+
+
+ML {* val card1_suc_f = Thm.freezeT (atomize_thm @{thm card1_suc}) *}
+
+prove card1_suc_r: {*
+ Logic.mk_implies
+   ((prop_of card1_suc_f),
+   (regularise (prop_of card1_suc_f) @{typ "'a List.list"} @{term "op \<approx>"} @{context})) *}
+  apply (simp add: equiv_res_forall[OF equiv_list_eq] equiv_res_exists[OF equiv_list_eq])
+  done
+
+ML {* @{thm card1_suc_r} OF [card1_suc_f] *}
+
+ML {* val li = Thm.freezeT (atomize_thm @{thm fold1.simps(2)}) *}
+prove fold1_def_2_r: {*
+ Logic.mk_implies
+   ((prop_of li),
+   (regularise (prop_of li) @{typ "'a List.list"} @{term "op \<approx>"} @{context})) *}
+  apply (simp add: equiv_res_forall[OF equiv_list_eq])
+  done
+
+ML {* @{thm fold1_def_2_r} OF [li] *}
+
+
+lemma yy:
+  shows "(False = x memb []) = (False = IN (x::nat) EMPTY)"
+unfolding IN_def EMPTY_def
+apply(rule_tac f="(op =) False" in arg_cong)
+apply(rule mem_respects)
+apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)
+apply(rule list_eq.intros)
+done
+
+lemma
+  shows "IN (x::nat) EMPTY = False"
+using m1
+apply -
+apply(rule yy[THEN iffD1, symmetric])
+apply(simp)
+done
+
+lemma
+  shows "((x=y) \<or> (IN x xs) = (IN (x::nat) (INSERT y xs))) =
+         ((x=y) \<or> x memb REP_fset xs = x memb (y # REP_fset xs))"
+unfolding IN_def INSERT_def
+apply(rule_tac f="(op \<or>) (x=y)" in arg_cong)
+apply(rule_tac f="(op =) (x memb REP_fset xs)" in arg_cong)
+apply(rule mem_respects)
+apply(rule list_eq.intros(3))
+apply(unfold REP_fset_def ABS_fset_def)
+apply(simp only: QUOT_TYPE.REP_ABS_rsp[OF QUOT_TYPE_fset])
+apply(rule list_eq_refl)
+done
+
+lemma yyy:
+  shows "
+    (
+     (UNION EMPTY s = s) &
+     ((UNION (INSERT e s1) s2) = (INSERT e (UNION s1 s2)))
+    ) = (
+     ((ABS_fset ([] @ REP_fset s)) = s) &
+     ((ABS_fset ((e # (REP_fset s1)) @ REP_fset s2)) = ABS_fset (e # (REP_fset s1 @ REP_fset s2)))
+    )"
+  unfolding UNION_def EMPTY_def INSERT_def
+  apply(rule_tac f="(op &)" in arg_cong2)
+  apply(rule_tac f="(op =)" in arg_cong2)
+  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])
+  apply(rule append_respects_fst)
+  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)
+  apply(rule list_eq_refl)
+  apply(simp)
+  apply(rule_tac f="(op =)" in arg_cong2)
+  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])
+  apply(rule append_respects_fst)
+  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)
+  apply(rule list_eq_refl)
+  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])
+  apply(rule list_eq.intros(5))
+  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)
+  apply(rule list_eq_refl)
+done
+
+lemma
+  shows "
+     (UNION EMPTY s = s) &
+     ((UNION (INSERT e s1) s2) = (INSERT e (UNION s1 s2)))"
+  apply (simp add: yyy)
+  apply (simp add: QUOT_TYPE_I_fset.thm10)
+  done
+
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm m2}))
+*}
+
+ML {*
+cterm_of @{theory} (prop_of m1_novars);
+cterm_of @{theory} (build_repabs_term @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"});
+*}
+
+
+(* Has all the theorems about fset plugged in. These should be parameters to the tactic *)
+ML {*
+  fun transconv_fset_tac' ctxt =
+    (LocalDefs.unfold_tac @{context} fset_defs) THEN
+    ObjectLogic.full_atomize_tac 1 THEN
+    REPEAT_ALL_NEW (FIRST' [
+      rtac @{thm list_eq_refl},
+      rtac @{thm cons_preserves},
+      rtac @{thm mem_respects},
+      rtac @{thm card1_rsp},
+      rtac @{thm QUOT_TYPE_I_fset.R_trans2},
+      CHANGED o (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms QUOT_TYPE_I_fset.REP_ABS_rsp})),
+      Cong_Tac.cong_tac @{thm cong},
+      rtac @{thm ext}
+    ]) 1
+*}
+
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm m1}))
+  val goal = build_repabs_goal @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"}
+  val cgoal = cterm_of @{theory} goal
+  val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite false fset_defs_sym cgoal)
+*}
+
+(*notation ( output) "prop" ("#_" [1000] 1000) *)
+notation ( output) "Trueprop" ("#_" [1000] 1000)
+
+
+prove {* (Thm.term_of cgoal2) *}
+  apply (tactic {* transconv_fset_tac' @{context} *})
+  done
+
+thm length_append (* Not true but worth checking that the goal is correct *)
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm length_append}))
+  val goal = build_repabs_goal @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"}
+  val cgoal = cterm_of @{theory} goal
+  val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite false fset_defs_sym cgoal)
+*}
+prove {* Thm.term_of cgoal2 *}
+  apply (tactic {* transconv_fset_tac' @{context} *})
+  sorry
+
+thm m2
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm m2}))
+  val goal = build_repabs_goal @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"}
+  val cgoal = cterm_of @{theory} goal
+  val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite false fset_defs_sym cgoal)
+*}
+prove {* Thm.term_of cgoal2 *}
+  apply (tactic {* transconv_fset_tac' @{context} *})
+  done
+
+thm list_eq.intros(4)
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm list_eq.intros(4)}))
+  val goal = build_repabs_goal @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"}
+  val cgoal = cterm_of @{theory} goal
+  val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite false fset_defs_sym cgoal)
+  val cgoal3 = Thm.rhs_of (MetaSimplifier.rewrite true @{thms QUOT_TYPE_I_fset.thm10} cgoal2)
+*}
+
+(* It is the same, but we need a name for it. *)
+prove zzz : {* Thm.term_of cgoal3 *}
+  apply (tactic {* transconv_fset_tac' @{context} *})
+  done
+
+(*lemma zzz' :
+  "(REP_fset (INSERT a (INSERT a (ABS_fset xs))) \<approx> REP_fset (INSERT a (ABS_fset xs)))"
+  using list_eq.intros(4) by (simp only: zzz)
+
+thm QUOT_TYPE_I_fset.REPS_same
+ML {* val zzz'' = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same} @{thm zzz'} *}
+*)
+
+thm list_eq.intros(5)
+(* prove {* build_repabs_goal @{context} (atomize_thm @{thm list_eq.intros(5)}) consts @{typ "'a list"} @{typ "'a fset"} *} *)
+ML {*
+  val m1_novars = snd(no_vars ((Context.Theory @{theory}), @{thm list_eq.intros(5)}))
+  val goal = build_repabs_goal @{context} m1_novars consts @{typ "'a list"} @{typ "'a fset"}
+  val cgoal = cterm_of @{theory} goal
+  val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite true fset_defs_sym cgoal)
+*}
+prove {* Thm.term_of cgoal2 *}
+  apply (tactic {* transconv_fset_tac' @{context} *})
+  done
+
+ML {*
+  fun lift_theorem_fset_aux thm lthy =
+    let
+      val ((_, [novars]), lthy2) = Variable.import true [thm] lthy;
+      val goal = build_repabs_goal @{context} novars consts @{typ "'a list"} @{typ "'a fset"};
+      val cgoal = cterm_of @{theory} goal;
+      val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite true fset_defs_sym cgoal);
+      val tac = transconv_fset_tac' @{context};
+      val cthm = Goal.prove_internal [] cgoal2 (fn _ => tac);
+      val nthm = MetaSimplifier.rewrite_rule [symmetric cthm] (snd (no_vars (Context.Theory @{theory}, thm)))
+      val nthm2 = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same QUOT_TYPE_I_fset.thm10} nthm;
+      val [nthm3] = ProofContext.export lthy2 lthy [nthm2]
+    in
+      nthm3
+    end
+*}
+
+ML {* lift_theorem_fset_aux @{thm m1} @{context} *}
+
+ML {*
+  fun lift_theorem_fset name thm lthy =
+    let
+      val lifted_thm = lift_theorem_fset_aux thm lthy;
+      val (_, lthy2) = note (name, lifted_thm) lthy;
+    in
+      lthy2
+    end;
+*}
+
+(* These do not work without proper definitions to rewrite back *)
+local_setup {* lift_theorem_fset @{binding "m1_lift"} @{thm m1} *}
+local_setup {* lift_theorem_fset @{binding "leqi4_lift"} @{thm list_eq.intros(4)} *}
+local_setup {* lift_theorem_fset @{binding "leqi5_lift"} @{thm list_eq.intros(5)} *}
+local_setup {* lift_theorem_fset @{binding "m2_lift"} @{thm m2} *}
+thm m1_lift
+thm leqi4_lift
+thm leqi5_lift
+thm m2_lift
+ML {* @{thm card1_suc_r} OF [card1_suc_f] *}
+(*ML {* Toplevel.program (fn () => lift_theorem_fset @{binding "card_suc"}
+     (@{thm card1_suc_r} OF [card1_suc_f]) @{context}) *}*)
+(*local_setup {* lift_theorem_fset @{binding "card_suc"} @{thm card1_suc} *}*)
+
+thm leqi4_lift
+ML {*
+  val (nam, typ) = hd (Term.add_vars (prop_of @{thm leqi4_lift}) [])
+  val (_, l) = dest_Type typ
+  val t = Type ("FSet.fset", l)
+  val v = Var (nam, t)
+  val cv = cterm_of @{theory} ((term_of @{cpat "REP_fset"}) $ v)
+*}
+
+ML {*
+  Toplevel.program (fn () =>
+    MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.thm10} (
+      Drule.instantiate' [] [NONE, SOME (cv)] @{thm leqi4_lift}
+    )
+  )
+*}
+
+
+
+(*prove aaa: {* (Thm.term_of cgoal2) *}
+  apply (tactic {* LocalDefs.unfold_tac @{context} fset_defs *} )
+  apply (atomize(full))
+  apply (tactic {* transconv_fset_tac' @{context} 1 *})
+  done*)
+
+(*
+datatype obj1 =
+  OVAR1 "string"
+| OBJ1 "(string * (string \<Rightarrow> obj1)) list"
+| INVOKE1 "obj1 \<Rightarrow> string"
+| UPDATE1 "obj1 \<Rightarrow> string \<Rightarrow> (string \<Rightarrow> obj1)"
+*)
+
+end




