src/HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML

   val sum_prod_thms_map: thm list
   val sum_prod_thms_set: thm list
   val sum_prod_thms_rel: thm list
 
   val mk_coinduct_tac: Proof.context -> thm list -> int -> int list -> thm -> thm list ->
-    thm list -> thm list -> thm list list -> thm list list list -> thm list list list -> tactic
-  val mk_coiter_tac: thm list -> thm list -> thm -> thm -> thm -> Proof.context -> tactic
+    thm list -> thm list -> thm list -> thm list -> thm list list -> thm list list list ->
+    thm list list list -> tactic
+  val mk_coiter_tac: thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context -> tactic
   val mk_ctor_iff_dtor_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
     tactic
   val mk_disc_coiter_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
   val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
-  val mk_half_distinct_tac: Proof.context -> thm -> thm list -> tactic
+  val mk_half_distinct_tac: Proof.context -> thm -> thm -> thm list -> tactic
   val mk_induct_tac: Proof.context -> int -> int list -> int list list -> int list list list ->
-    thm list -> thm -> thm list -> thm list list -> tactic
-  val mk_inject_tac: Proof.context -> thm -> thm -> tactic
-  val mk_iter_tac: thm list -> thm list -> thm list -> thm -> thm -> Proof.context -> tactic
+    thm list -> thm -> thm list -> thm list -> thm list -> thm list list -> tactic
+  val mk_inject_tac: Proof.context -> thm -> thm -> thm -> tactic
+  val mk_iter_tac: thm list -> thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context ->
+    tactic
 end;
 
 structure BNF_FP_Def_Sugar_Tactics : BNF_FP_DEF_SUGAR_TACTICS =
 struct
 

     EVERY' (map3 (fn cTs => fn cx => fn th =>
       dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
       SELECT_GOAL (unfold_thms_tac ctxt [th]) THEN'
       atac) [rev cTs, cTs] [cdtor, cctor] [dtor_ctor, ctor_dtor]));
 
-fun mk_half_distinct_tac ctxt ctor_inject ctr_defs =
-  unfold_thms_tac ctxt (ctor_inject :: @{thms sum.inject} @ ctr_defs) THEN
+fun mk_half_distinct_tac ctxt ctor_inject abs_inject ctr_defs =
+  unfold_thms_tac ctxt (ctor_inject :: abs_inject :: @{thms sum.inject} @ ctr_defs) THEN
   HEADGOAL (rtac @{thm sum.distinct(1)});
 
-fun mk_inject_tac ctxt ctr_def ctor_inject =
-  unfold_thms_tac ctxt [ctr_def] THEN HEADGOAL (rtac (ctor_inject RS ssubst)) THEN
-  unfold_thms_tac ctxt @{thms sum.inject Pair_eq conj_assoc} THEN HEADGOAL (rtac refl);
+fun mk_inject_tac ctxt ctr_def ctor_inject abs_inject =
+  unfold_thms_tac ctxt [ctr_def] THEN
+  HEADGOAL (rtac (ctor_inject RS ssubst)) THEN
+  unfold_thms_tac ctxt (abs_inject :: @{thms sum.inject Pair_eq conj_assoc}) THEN
+  HEADGOAL (rtac refl);
 
 val iter_unfold_thms =
   @{thms comp_def convol_def fst_conv id_def case_prod_Pair_iden snd_conv split_conv
       case_unit_Unity} @ sum_prod_thms_map;
 
-fun mk_iter_tac pre_map_defs map_idents iter_defs ctor_iter ctr_def ctxt =
-  unfold_thms_tac ctxt (ctr_def :: ctor_iter :: iter_defs @ pre_map_defs @ map_idents @
-    iter_unfold_thms) THEN HEADGOAL (rtac refl);
+fun mk_iter_tac pre_map_defs map_idents iter_defs ctor_iter fp_abs_inverse abs_inverse ctr_def ctxt =
+  unfold_thms_tac ctxt (ctr_def :: ctor_iter :: fp_abs_inverse :: abs_inverse :: iter_defs @
+    pre_map_defs @ map_idents @ iter_unfold_thms) THEN HEADGOAL (rtac refl);
 
 val coiter_unfold_thms = @{thms id_def} @ sum_prod_thms_map;
-val ss_if_True_False = simpset_of (ss_only @{thms if_True if_False} @{context});
 
-fun mk_coiter_tac coiter_defs map_idents ctor_dtor_coiter pre_map_def ctr_def ctxt =
-  unfold_thms_tac ctxt (ctr_def :: coiter_defs) THEN
-  HEADGOAL (rtac (ctor_dtor_coiter RS trans) THEN'
-    asm_simp_tac (put_simpset ss_if_True_False ctxt)) THEN_MAYBE
-  (unfold_thms_tac ctxt (pre_map_def :: map_idents @ coiter_unfold_thms) THEN
-   HEADGOAL (rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong)));
+fun mk_coiter_tac coiter_defs map_idents ctor_dtor_coiter pre_map_def abs_inverse ctr_def ctxt =
+  let
+    val ss = ss_only (pre_map_def :: abs_inverse :: map_idents @ coiter_unfold_thms @
+      @{thms o_apply vimage2p_def if_True if_False}) ctxt;
+  in
+    unfold_thms_tac ctxt (ctr_def :: coiter_defs) THEN
+    HEADGOAL (rtac (ctor_dtor_coiter RS trans) THEN' asm_simp_tac ss) THEN_MAYBE
+    HEADGOAL (rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong))
+  end;
 
 fun mk_disc_coiter_iff_tac case_splits' coiters discs ctxt =
   EVERY (map3 (fn case_split_tac => fn coiter_thm => fn disc =>
       HEADGOAL case_split_tac THEN unfold_thms_tac ctxt [coiter_thm] THEN
       HEADGOAL (asm_simp_tac (ss_only basic_simp_thms ctxt)) THEN

 fun solve_prem_prem_tac ctxt =
   REPEAT o (eresolve_tac @{thms bexE rev_bexI} ORELSE' rtac @{thm rev_bexI[OF UNIV_I]} ORELSE'
     hyp_subst_tac ctxt ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'
   (rtac refl ORELSE' atac ORELSE' rtac @{thm singletonI});
 
-fun mk_induct_leverage_prem_prems_tac ctxt nn kks set_maps pre_set_defs =
+fun mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
+    pre_set_defs =
   HEADGOAL (EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
-    SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ set_maps @ sum_prod_thms_set0)),
+    SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ fp_abs_inverses @ abs_inverses @ set_maps @
+      sum_prod_thms_set0)),
     solve_prem_prem_tac ctxt]) (rev kks)));
 
-fun mk_induct_discharge_prem_tac ctxt nn n set_maps pre_set_defs m k kks =
+fun mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps pre_set_defs m k
+    kks =
   let val r = length kks in
     HEADGOAL (EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac ctxt,
       REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)]) THEN
     EVERY [REPEAT_DETERM_N r
         (HEADGOAL (rotate_tac ~1 THEN' dtac meta_mp THEN' rotate_tac 1) THEN prefer_tac 2),
       if r > 0 then ALLGOALS (Goal.norm_hhf_tac ctxt) else all_tac, HEADGOAL atac,
-      mk_induct_leverage_prem_prems_tac ctxt nn kks set_maps pre_set_defs]
+      mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
+        pre_set_defs]
   end;
 
-fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' set_maps pre_set_defss =
+fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' fp_abs_inverses abs_inverses set_maps
+    pre_set_defss =
   let val n = Integer.sum ns in
     unfold_thms_tac ctxt ctr_defs THEN HEADGOAL (rtac ctor_induct') THEN
     co_induct_inst_as_projs_tac ctxt 0 THEN
-    EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt nn n set_maps) pre_set_defss
-      mss (unflat mss (1 upto n)) kkss)
+    EVERY (map4 (EVERY oooo map3 o
+        mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps)
+      pre_set_defss mss (unflat mss (1 upto n)) kkss)
   end;
 
-fun mk_coinduct_same_ctr_tac ctxt rel_eqs pre_rel_def dtor_ctor ctr_def discs sels =
+fun mk_coinduct_same_ctr_tac ctxt rel_eqs pre_rel_def fp_abs_inverse abs_inverse dtor_ctor ctr_def
+    discs sels =
   hyp_subst_tac ctxt THEN'
   CONVERSION (hhf_concl_conv
     (Conv.top_conv (K (Conv.try_conv (Conv.rewr_conv ctr_def))) ctxt) ctxt) THEN'
   SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels)) THEN'
-  SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels @ sum_prod_thms_rel)) THEN'
+  SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: fp_abs_inverse :: abs_inverse :: dtor_ctor ::
+    sels @ sum_prod_thms_rel @ @{thms o_apply vimage2p_def})) THEN'
   (atac ORELSE' REPEAT o etac conjE THEN'
      full_simp_tac
        (ss_only (@{thm prod.inject} :: no_refl discs @ rel_eqs @ more_simp_thms) ctxt) THEN'
      REPEAT o etac conjE THEN_MAYBE' REPEAT o hyp_subst_tac ctxt THEN'
      REPEAT o (resolve_tac [refl, conjI] ORELSE' atac));

   in
     hyp_subst_tac ctxt THEN' REPEAT o etac conjE THEN'
     full_simp_tac (ss_only (refl :: no_refl discs'' @ basic_simp_thms) ctxt)
   end;
 
-fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def dtor_ctor exhaust ctr_defs
-    discss selss =
+fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def fp_abs_inverse abs_inverse
+    dtor_ctor exhaust ctr_defs discss selss =
   let val ks = 1 upto n in
     EVERY' ([rtac allI, rtac allI, rtac impI, select_prem_tac nn (dtac meta_spec) kk,
         dtac meta_spec, dtac meta_mp, atac, rtac exhaust, K (co_induct_inst_as_projs_tac ctxt 0),
         hyp_subst_tac ctxt] @
       map4 (fn k => fn ctr_def => fn discs => fn sels =>
         EVERY' ([rtac exhaust, K (co_induct_inst_as_projs_tac ctxt 1)] @
           map2 (fn k' => fn discs' =>
             if k' = k then
-              mk_coinduct_same_ctr_tac ctxt rel_eqs' pre_rel_def dtor_ctor ctr_def discs sels
+              mk_coinduct_same_ctr_tac ctxt rel_eqs' pre_rel_def fp_abs_inverse abs_inverse
+                dtor_ctor ctr_def discs sels
             else
               mk_coinduct_distinct_ctrs_tac ctxt discs discs') ks discss)) ks ctr_defs discss selss)
   end;
 
-fun mk_coinduct_tac ctxt rel_eqs' nn ns dtor_coinduct' pre_rel_defs dtor_ctors exhausts ctr_defss
-    discsss selsss =
+fun mk_coinduct_tac ctxt rel_eqs' nn ns dtor_coinduct' pre_rel_defs fp_abs_inverses abs_inverses
+    dtor_ctors exhausts ctr_defss discsss selsss =
   HEADGOAL (rtac dtor_coinduct' THEN'
-    EVERY' (map8 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
-      (1 upto nn) ns pre_rel_defs dtor_ctors exhausts ctr_defss discsss selsss));
+    EVERY' (map10 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
+      (1 upto nn) ns pre_rel_defs fp_abs_inverses abs_inverses dtor_ctors exhausts ctr_defss discsss
+      selsss));
 
 end;