(* Title: CCL/Trancl.ML ID: $Id: Trancl.ML,v 1.7 2005/09/17 15:35:29 wenzelm Exp $ *) (** Natural deduction for trans(r) **) val prems = goalw (the_context ()) [trans_def] "(!! x y z. [| <x,y>:r; <y,z>:r |] ==> <x,z>:r) ==> trans(r)"; by (REPEAT (ares_tac (prems@[allI,impI]) 1)); qed "transI"; val major::prems = goalw (the_context ()) [trans_def] "[| trans(r); <a,b>:r; <b,c>:r |] ==> <a,c>:r"; by (cut_facts_tac [major] 1); by (fast_tac (FOL_cs addIs prems) 1); qed "transD"; (** Identity relation **) Goalw [id_def] "<a,a> : id"; by (rtac CollectI 1); by (rtac exI 1); by (rtac refl 1); qed "idI"; val major::prems = goalw (the_context ()) [id_def] "[| p: id; !!x.[| p = <x,x> |] ==> P \ \ |] ==> P"; by (rtac (major RS CollectE) 1); by (etac exE 1); by (eresolve_tac prems 1); qed "idE"; (** Composition of two relations **) val prems = goalw (the_context ()) [comp_def] "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s"; by (fast_tac (set_cs addIs prems) 1); qed "compI"; (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*) val prems = goalw (the_context ()) [comp_def] "[| xz : r O s; \ \ !!x y z. [| xz = <x,z>; <x,y>:s; <y,z>:r |] ==> P \ \ |] ==> P"; by (cut_facts_tac prems 1); by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1)); qed "compE"; val prems = goal (the_context ()) "[| <a,c> : r O s; \ \ !!y. [| <a,y>:s; <y,c>:r |] ==> P \ \ |] ==> P"; by (rtac compE 1); by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [pair_inject,ssubst] 1)); qed "compEpair"; val comp_cs = set_cs addIs [compI,idI] addEs [compE,idE] addSEs [pair_inject]; val prems = goal (the_context ()) "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"; by (cut_facts_tac prems 1); by (fast_tac comp_cs 1); qed "comp_mono"; (** The relation rtrancl **) Goal "mono(%s. id Un (r O s))"; by (rtac monoI 1); by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1)); qed "rtrancl_fun_mono"; val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski); (*Reflexivity of rtrancl*) Goal "<a,a> : r^*"; by (stac rtrancl_unfold 1); by (fast_tac comp_cs 1); qed "rtrancl_refl"; (*Closure under composition with r*) val prems = goal (the_context ()) "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^*"; by (stac rtrancl_unfold 1); by (fast_tac (comp_cs addIs prems) 1); qed "rtrancl_into_rtrancl"; (*rtrancl of r contains r*) val [prem] = goal (the_context ()) "[| <a,b> : r |] ==> <a,b> : r^*"; by (rtac (rtrancl_refl RS rtrancl_into_rtrancl) 1); by (rtac prem 1); qed "r_into_rtrancl"; (** standard induction rule **) val major::prems = goal (the_context ()) "[| <a,b> : r^*; \ \ !!x. P(<x,x>); \ \ !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |] ==> P(<x,z>) |] \ \ ==> P(<a,b>)"; by (rtac (major RS (rtrancl_def RS def_induct)) 1); by (rtac rtrancl_fun_mono 1); by (fast_tac (comp_cs addIs prems) 1); qed "rtrancl_full_induct"; (*nice induction rule*) val major::prems = goal (the_context ()) "[| <a,b> : r^*; \ \ P(a); \ \ !!y z.[| <a,y> : r^*; <y,z> : r; P(y) |] ==> P(z) |] \ \ ==> P(b)"; (*by induction on this formula*) by (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)" 1); (*now solve first subgoal: this formula is sufficient*) by (fast_tac FOL_cs 1); (*now do the induction*) by (resolve_tac [major RS rtrancl_full_induct] 1); by (fast_tac (comp_cs addIs prems) 1); by (fast_tac (comp_cs addIs prems) 1); qed "rtrancl_induct"; (*transitivity of transitive closure!! -- by induction.*) Goal "trans(r^*)"; by (rtac transI 1); by (res_inst_tac [("b","z")] rtrancl_induct 1); by (DEPTH_SOLVE (eresolve_tac [asm_rl, rtrancl_into_rtrancl] 1)); qed "trans_rtrancl"; (*elimination of rtrancl -- by induction on a special formula*) val major::prems = goal (the_context ()) "[| <a,b> : r^*; (a = b) ==> P; \ \ !!y.[| <a,y> : r^*; <y,b> : r |] ==> P |] \ \ ==> P"; by (subgoal_tac "a = b | (EX y. <a,y> : r^* & <y,b> : r)" 1); by (rtac (major RS rtrancl_induct) 2); by (fast_tac (set_cs addIs prems) 2); by (fast_tac (set_cs addIs prems) 2); by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1)); qed "rtranclE"; (**** The relation trancl ****) (** Conversions between trancl and rtrancl **) val [major] = goalw (the_context ()) [trancl_def] "[| <a,b> : r^+ |] ==> <a,b> : r^*"; by (resolve_tac [major RS compEpair] 1); by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1)); qed "trancl_into_rtrancl"; (*r^+ contains r*) val [prem] = goalw (the_context ()) [trancl_def] "[| <a,b> : r |] ==> <a,b> : r^+"; by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1)); qed "r_into_trancl"; (*intro rule by definition: from rtrancl and r*) val prems = goalw (the_context ()) [trancl_def] "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^+"; by (REPEAT (resolve_tac ([compI]@prems) 1)); qed "rtrancl_into_trancl1"; (*intro rule from r and rtrancl*) val prems = goal (the_context ()) "[| <a,b> : r; <b,c> : r^* |] ==> <a,c> : r^+"; by (resolve_tac (prems RL [rtranclE]) 1); by (etac subst 1); by (resolve_tac (prems RL [r_into_trancl]) 1); by (rtac (trans_rtrancl RS transD RS rtrancl_into_trancl1) 1); by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 1)); qed "rtrancl_into_trancl2"; (*elimination of r^+ -- NOT an induction rule*) val major::prems = goal (the_context ()) "[| <a,b> : r^+; \ \ <a,b> : r ==> P; \ \ !!y.[| <a,y> : r^+; <y,b> : r |] ==> P \ \ |] ==> P"; by (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ & <y,b> : r)" 1); by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1)); by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1); by (etac rtranclE 1); by (fast_tac comp_cs 1); by (fast_tac (comp_cs addSIs [rtrancl_into_trancl1]) 1); qed "tranclE"; (*Transitivity of r^+. Proved by unfolding since it uses transitivity of rtrancl. *) Goalw [trancl_def] "trans(r^+)"; by (rtac transI 1); by (REPEAT (etac compEpair 1)); by (rtac (rtrancl_into_rtrancl RS (trans_rtrancl RS transD RS compI)) 1); by (REPEAT (assume_tac 1)); qed "trans_trancl"; val prems = goal (the_context ()) "[| <a,b> : r; <b,c> : r^+ |] ==> <a,c> : r^+"; by (rtac (r_into_trancl RS (trans_trancl RS transD)) 1); by (resolve_tac prems 1); by (resolve_tac prems 1); qed "trancl_into_trancl2";