1 (* Title: HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML
2 Author: Jasmin Blanchette, TU Muenchen
5 Tactics for datatype and codatatype sugar.
8 signature BNF_FP_DEF_SUGAR_TACTICS =
10 val sumprod_thms_map: thm list
11 val sumprod_thms_set: thm list
12 val sumprod_thms_rel: thm list
14 val mk_coinduct_tac: Proof.context -> thm list -> int -> int list -> thm -> thm list ->
15 thm list -> thm list -> thm list -> thm list -> thm list list -> thm list list list ->
16 thm list list list -> tactic
17 val mk_corec_tac: thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context -> tactic
18 val mk_ctor_iff_dtor_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
20 val mk_disc_corec_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
21 val mk_disc_map_iff_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> tactic
22 val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
23 val mk_half_distinct_tac: Proof.context -> thm -> thm -> thm list -> tactic
24 val mk_induct_tac: Proof.context -> int -> int list -> int list list -> int list list list ->
25 thm list -> thm -> thm list -> thm list -> thm list -> thm list list -> tactic
26 val mk_inject_tac: Proof.context -> thm -> thm -> thm -> tactic
27 val mk_rec_tac: thm list -> thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context ->
29 val mk_rel_coinduct0_tac: Proof.context -> thm -> cterm list -> thm list -> thm list ->
30 thm list list -> thm list list -> thm list list -> thm list -> thm list -> thm list ->
31 thm list -> thm list -> thm list -> tactic
32 val mk_rel_induct0_tac: Proof.context -> thm -> thm list -> cterm list -> thm list ->
33 thm list list -> thm list -> thm list -> thm list -> thm list -> tactic
34 val mk_sel_map_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> thm list -> tactic
35 val mk_sel_set_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> thm list -> tactic
36 val mk_set_empty_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> tactic
39 structure BNF_FP_Def_Sugar_Tactics : BNF_FP_DEF_SUGAR_TACTICS =
47 val basic_simp_thms = @{thms simp_thms(7,8,12,14,22,24)};
48 val more_simp_thms = basic_simp_thms @ @{thms simp_thms(11,15,16,21)};
49 val simp_thms' = @{thms simp_thms(6,7,8,11,12,15,16,22,24)};
51 val sumprod_thms_map = @{thms id_apply map_prod_simp prod.case sum.case map_sum.simps};
52 val sumprod_thms_set =
53 @{thms UN_empty UN_insert Un_empty_left Un_empty_right Un_iff UN_simps(10) UN_iff
54 Union_Un_distrib image_iff o_apply map_prod_simp
55 mem_Collect_eq prod_set_simps map_sum.simps sum_set_simps};
56 val sumprod_thms_rel = @{thms rel_sum_simps rel_prod_apply prod.inject id_apply conj_assoc};
58 fun hhf_concl_conv cv ctxt ct =
59 (case Thm.term_of ct of
60 Const (@{const_name Pure.all}, _) $ Abs _ =>
61 Conv.arg_conv (Conv.abs_conv (hhf_concl_conv cv o snd) ctxt) ct
62 | _ => Conv.concl_conv ~1 cv ct);
64 fun co_induct_inst_as_projs ctxt k thm =
66 val fs = Term.add_vars (prop_of thm) []
67 |> filter (fn (_, Type (@{type_name fun}, [_, T'])) => T' <> HOLogic.boolT | _ => false);
68 fun mk_cfp (f as (_, T)) =
69 (certify ctxt (Var f), certify ctxt (mk_proj T (num_binder_types T) k));
70 val cfps = map mk_cfp fs;
72 Drule.cterm_instantiate cfps thm
75 val co_induct_inst_as_projs_tac = PRIMITIVE oo co_induct_inst_as_projs;
77 fun mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor sumEN' =
78 unfold_thms_tac ctxt (ctor_iff_dtor :: ctr_defs) THEN HEADGOAL (rtac sumEN') THEN
79 HEADGOAL (EVERY' (maps (fn k => [select_prem_tac n (rotate_tac 1) k,
80 REPEAT_DETERM o dtac meta_spec, etac meta_mp, atac]) (1 upto n)));
82 fun mk_ctor_iff_dtor_tac ctxt cTs cctor cdtor ctor_dtor dtor_ctor =
83 HEADGOAL (rtac iffI THEN'
84 EVERY' (map3 (fn cTs => fn cx => fn th =>
85 dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
86 SELECT_GOAL (unfold_thms_tac ctxt [th]) THEN'
87 atac) [rev cTs, cTs] [cdtor, cctor] [dtor_ctor, ctor_dtor]));
89 fun mk_half_distinct_tac ctxt ctor_inject abs_inject ctr_defs =
90 unfold_thms_tac ctxt (ctor_inject :: abs_inject :: @{thms sum.inject} @ ctr_defs) THEN
91 HEADGOAL (rtac @{thm sum.distinct(1)});
93 fun mk_inject_tac ctxt ctr_def ctor_inject abs_inject =
94 unfold_thms_tac ctxt [ctr_def] THEN
95 HEADGOAL (rtac (ctor_inject RS ssubst)) THEN
96 unfold_thms_tac ctxt (abs_inject :: @{thms sum.inject prod.inject conj_assoc}) THEN
100 @{thms comp_def convol_def fst_conv id_def case_prod_Pair_iden snd_conv split_conv
101 case_unit_Unity} @ sumprod_thms_map;
103 fun mk_rec_tac pre_map_defs map_ident0s rec_defs ctor_rec fp_abs_inverse abs_inverse ctr_def ctxt =
104 unfold_thms_tac ctxt (ctr_def :: ctor_rec :: fp_abs_inverse :: abs_inverse :: rec_defs @
105 pre_map_defs @ map_ident0s @ rec_unfold_thms) THEN HEADGOAL (rtac refl);
107 val corec_unfold_thms = @{thms id_def} @ sumprod_thms_map;
109 fun mk_corec_tac corec_defs map_ident0s ctor_dtor_corec pre_map_def abs_inverse ctr_def ctxt =
111 val ss = ss_only (pre_map_def :: abs_inverse :: map_ident0s @ corec_unfold_thms @
112 @{thms o_apply vimage2p_def if_True if_False}) ctxt;
114 unfold_thms_tac ctxt (ctr_def :: corec_defs) THEN
115 HEADGOAL (rtac (ctor_dtor_corec RS trans) THEN' asm_simp_tac ss) THEN_MAYBE
116 HEADGOAL (rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong))
119 fun mk_disc_corec_iff_tac case_splits' corecs discs ctxt =
120 EVERY (map3 (fn case_split_tac => fn corec_thm => fn disc =>
121 HEADGOAL case_split_tac THEN unfold_thms_tac ctxt [corec_thm] THEN
122 HEADGOAL (asm_simp_tac (ss_only basic_simp_thms ctxt)) THEN
123 (if is_refl disc then all_tac else HEADGOAL (rtac disc)))
124 (map rtac case_splits' @ [K all_tac]) corecs discs);
126 fun mk_disc_map_iff_tac ctxt ct exhaust discs maps =
127 TRYALL Goal.conjunction_tac THEN
128 ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
129 REPEAT_DETERM o hyp_subst_tac ctxt) THEN
130 unfold_thms_tac ctxt maps THEN
131 unfold_thms_tac ctxt (map (fn thm => thm RS @{thm iffD2[OF eq_False]}
132 handle THM _ => thm RS @{thm iffD2[OF eq_True]}) discs) THEN
133 ALLGOALS (rtac refl ORELSE' rtac TrueI);
135 fun solve_prem_prem_tac ctxt =
136 REPEAT o (eresolve_tac @{thms bexE rev_bexI} ORELSE' rtac @{thm rev_bexI[OF UNIV_I]} ORELSE'
137 hyp_subst_tac ctxt ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'
138 (rtac refl ORELSE' atac ORELSE' rtac @{thm singletonI});
140 fun mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
142 HEADGOAL (EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
143 SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ fp_abs_inverses @ abs_inverses @ set_maps @
145 solve_prem_prem_tac ctxt]) (rev kks)));
147 fun mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps pre_set_defs m k
149 let val r = length kks in
150 HEADGOAL (EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac ctxt,
151 REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)]) THEN
152 EVERY [REPEAT_DETERM_N r
153 (HEADGOAL (rotate_tac ~1 THEN' dtac meta_mp THEN' rotate_tac 1) THEN prefer_tac 2),
154 if r > 0 then ALLGOALS (Goal.norm_hhf_tac ctxt) else all_tac, HEADGOAL atac,
155 mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
159 fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' fp_abs_inverses abs_inverses set_maps
161 let val n = Integer.sum ns in
162 unfold_thms_tac ctxt ctr_defs THEN HEADGOAL (rtac ctor_induct') THEN
163 co_induct_inst_as_projs_tac ctxt 0 THEN
164 EVERY (map4 (EVERY oooo map3 o
165 mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps)
166 pre_set_defss mss (unflat mss (1 upto n)) kkss)
169 fun mk_coinduct_same_ctr_tac ctxt rel_eqs pre_rel_def fp_abs_inverse abs_inverse dtor_ctor ctr_def
171 hyp_subst_tac ctxt THEN'
172 CONVERSION (hhf_concl_conv
173 (Conv.top_conv (K (Conv.try_conv (Conv.rewr_conv ctr_def))) ctxt) ctxt) THEN'
174 SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels)) THEN'
175 SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: fp_abs_inverse :: abs_inverse :: dtor_ctor ::
176 sels @ sumprod_thms_rel @ @{thms o_apply vimage2p_def})) THEN'
177 (atac ORELSE' REPEAT o etac conjE THEN'
178 full_simp_tac (ss_only (no_refl discs @ rel_eqs @ more_simp_thms) ctxt) THEN'
179 REPEAT o etac conjE THEN_MAYBE' REPEAT o hyp_subst_tac ctxt THEN'
180 REPEAT o (resolve_tac [refl, conjI] ORELSE' atac));
182 fun mk_coinduct_distinct_ctrs_tac ctxt discs discs' =
184 val discs'' = map (perhaps (try (fn th => th RS @{thm notnotD}))) (discs @ discs')
185 |> distinct Thm.eq_thm_prop;
187 hyp_subst_tac ctxt THEN' REPEAT o etac conjE THEN'
188 full_simp_tac (ss_only (refl :: no_refl discs'' @ basic_simp_thms) ctxt)
191 fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def fp_abs_inverse abs_inverse
192 dtor_ctor exhaust ctr_defs discss selss =
193 let val ks = 1 upto n in
194 EVERY' ([rtac allI, rtac allI, rtac impI, select_prem_tac nn (dtac meta_spec) kk,
195 dtac meta_spec, dtac meta_mp, atac, rtac exhaust, K (co_induct_inst_as_projs_tac ctxt 0),
196 hyp_subst_tac ctxt] @
197 map4 (fn k => fn ctr_def => fn discs => fn sels =>
198 EVERY' ([rtac exhaust, K (co_induct_inst_as_projs_tac ctxt 1)] @
199 map2 (fn k' => fn discs' =>
201 mk_coinduct_same_ctr_tac ctxt rel_eqs' pre_rel_def fp_abs_inverse abs_inverse
202 dtor_ctor ctr_def discs sels
204 mk_coinduct_distinct_ctrs_tac ctxt discs discs') ks discss)) ks ctr_defs discss selss)
207 fun mk_coinduct_tac ctxt rel_eqs' nn ns dtor_coinduct' pre_rel_defs fp_abs_inverses abs_inverses
208 dtor_ctors exhausts ctr_defss discsss selsss =
209 HEADGOAL (rtac dtor_coinduct' THEN'
210 EVERY' (map10 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
211 (1 upto nn) ns pre_rel_defs fp_abs_inverses abs_inverses dtor_ctors exhausts ctr_defss discsss
214 fun mk_rel_coinduct0_tac ctxt dtor_rel_coinduct cts assms exhausts discss selss ctor_defss
215 dtor_ctors ctor_injects abs_injects rel_pre_defs abs_inverses nesting_rel_eqs =
216 rtac dtor_rel_coinduct 1 THEN
217 EVERY (map11 (fn ct => fn assm => fn exhaust => fn discs => fn sels => fn ctor_defs =>
218 fn dtor_ctor => fn ctor_inject => fn abs_inject => fn rel_pre_def => fn abs_inverse =>
219 (rtac exhaust THEN_ALL_NEW (rtac exhaust THEN_ALL_NEW
220 (dtac (rotate_prems (~1) ((cterm_instantiate_pos [NONE, NONE, NONE, NONE, SOME ct] @{thm
221 arg_cong2}) RS iffD1)) THEN'
222 atac THEN' atac THEN' hyp_subst_tac ctxt THEN' dtac assm THEN'
223 REPEAT_DETERM o etac conjE))) 1 THEN
224 Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @ sels
226 Local_Defs.unfold_tac ctxt (dtor_ctor :: rel_pre_def :: abs_inverse :: ctor_inject ::
227 abs_inject :: ctor_defs @ nesting_rel_eqs @ simp_thms' @ @{thms BNF_Comp.id_bnf_comp_def
228 rel_sum_simps rel_prod_apply vimage2p_def Inl_Inr_False iffD2[OF eq_False Inr_not_Inl]
229 sum.inject prod.inject}) THEN
230 REPEAT_DETERM (HEADGOAL ((REPEAT_DETERM o etac conjE) THEN' (REPEAT_DETERM o rtac conjI) THEN'
231 (rtac refl ORELSE' atac))))
232 cts assms exhausts discss selss ctor_defss dtor_ctors ctor_injects abs_injects rel_pre_defs
235 fun mk_rel_induct0_tac ctxt ctor_rel_induct assms cterms exhausts ctor_defss ctor_injects
236 rel_pre_list_defs Abs_inverses nesting_rel_eqs =
237 rtac ctor_rel_induct 1 THEN EVERY (map6 (fn cterm => fn exhaust => fn ctor_defs =>
238 fn ctor_inject => fn rel_pre_list_def => fn Abs_inverse =>
239 HEADGOAL (rtac exhaust THEN_ALL_NEW (rtac exhaust THEN_ALL_NEW
240 (rtac (cterm_instantiate_pos (replicate 4 NONE @ [SOME cterm]) @{thm arg_cong2} RS iffD2)
241 THEN' atac THEN' atac THEN' TRY o resolve_tac assms))) THEN
242 unfold_tac ctxt (ctor_inject :: rel_pre_list_def :: ctor_defs @ nesting_rel_eqs @
243 @{thms BNF_Comp.id_bnf_comp_def vimage2p_def}) THEN
244 TRYALL (hyp_subst_tac ctxt) THEN
245 unfold_tac ctxt (Abs_inverse :: @{thms rel_sum_simps rel_prod_apply Inl_Inr_False
246 Inr_Inl_False sum.inject prod.inject}) THEN
247 TRYALL (etac FalseE ORELSE' (REPEAT_DETERM o etac conjE) THEN' atac))
248 cterms exhausts ctor_defss ctor_injects rel_pre_list_defs Abs_inverses);
250 fun mk_sel_map_tac ctxt ct exhaust discs maps sels =
251 TRYALL Goal.conjunction_tac THEN
252 ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
253 REPEAT_DETERM o hyp_subst_tac ctxt) THEN
254 Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @
255 @{thms not_True_eq_False not_False_eq_True}) THEN
256 TRYALL (etac FalseE ORELSE' etac @{thm TrueE}) THEN
257 Local_Defs.unfold_tac ctxt (maps @ sels) THEN
258 ALLGOALS (rtac refl);
260 fun mk_sel_set_tac ctxt ct exhaust discs sels sets =
261 TRYALL Goal.conjunction_tac THEN
262 ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
263 REPEAT_DETERM o hyp_subst_tac ctxt) THEN
264 Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @
265 @{thms not_True_eq_False not_False_eq_True}) THEN
266 TRYALL (etac FalseE ORELSE' etac @{thm TrueE}) THEN
267 Local_Defs.unfold_tac ctxt (sels @ sets) THEN
269 REPEAT o (resolve_tac @{thms UnI1 UnI2 imageI} ORELSE'
270 eresolve_tac @{thms UN_I UN_I[rotated] imageE} ORELSE'
271 hyp_subst_tac ctxt) THEN'
272 (rtac @{thm singletonI} ORELSE' atac));
274 fun mk_set_empty_tac ctxt ct exhaust sets discs =
275 TRYALL Goal.conjunction_tac THEN
276 ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
277 REPEAT_DETERM o hyp_subst_tac ctxt) THEN
278 unfold_thms_tac ctxt (sets @ map_filter (fn thm =>
279 SOME (thm RS @{thm iffD2[OF eq_False]}) handle THM _ => NONE) discs) THEN
280 ALLGOALS (rtac refl ORELSE' etac FalseE);