diff --git a/thys/Saturation_Framework/Labeled_Lifting_to_Non_Ground_Calculi.thy b/thys/Saturation_Framework/Labeled_Lifting_to_Non_Ground_Calculi.thy
--- a/thys/Saturation_Framework/Labeled_Lifting_to_Non_Ground_Calculi.thy
+++ b/thys/Saturation_Framework/Labeled_Lifting_to_Non_Ground_Calculi.thy
@@ -1,438 +1,436 @@
 (*  Title:       Labeled Lifting to Non-Ground Calculi of the Saturation Framework
  *  Author:      Sophie Tourret <stourret at mpi-inf.mpg.de>, 2019-2020 *)
 
 section \<open>Labeled Liftings\<close>
 
 text \<open>This section formalizes the extension of the lifting results to labeled
   calculi. This corresponds to section 3.4 of the report.\<close>
 
 theory Labeled_Lifting_to_Non_Ground_Calculi
   imports Lifting_to_Non_Ground_Calculi
 begin
 
 subsection \<open>Labeled Lifting with a Family of Well-founded Orderings\<close>
 
 locale labeled_lifting_w_wf_ord_family =
   lifting_with_wf_ordering_family Bot_F Inf_F Bot_G entails_G Inf_G Red_Inf_G Red_F_G \<G>_F \<G>_Inf Prec_F
   for
     Bot_F :: "'f set" and
     Inf_F :: "'f inference set" and
     Bot_G :: "'g set" and
     entails_G :: "'g set \<Rightarrow> 'g set \<Rightarrow> bool"  (infix "\<Turnstile>G" 50) and
     Inf_G :: "'g inference set" and
     Red_Inf_G :: "'g set \<Rightarrow> 'g inference set" and
     Red_F_G :: "'g set \<Rightarrow> 'g set" and
     \<G>_F :: "'f \<Rightarrow> 'g set" and
     \<G>_Inf :: "'f inference \<Rightarrow> 'g inference set option" and
     Prec_F :: "'g \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
   + fixes
-    l :: \<open>'l itself\<close> and
     Inf_FL :: \<open>('f \<times> 'l) inference set\<close>
   assumes
     Inf_F_to_Inf_FL: \<open>\<iota>\<^sub>F \<in> Inf_F \<Longrightarrow> length (Ll :: 'l list) = length (prems_of \<iota>\<^sub>F) \<Longrightarrow>
       \<exists>L0. Infer (zip (prems_of \<iota>\<^sub>F) Ll) (concl_of \<iota>\<^sub>F, L0) \<in> Inf_FL\<close> and
     Inf_FL_to_Inf_F: \<open>\<iota>\<^sub>F\<^sub>L \<in> Inf_FL \<Longrightarrow> Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L)) \<in> Inf_F\<close>
 begin
 
 definition to_F :: \<open>('f \<times> 'l) inference \<Rightarrow> 'f inference\<close> where
   \<open>to_F \<iota>\<^sub>F\<^sub>L = Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L))\<close>
 
 definition Bot_FL :: \<open>('f \<times> 'l) set\<close> where \<open>Bot_FL = Bot_F \<times> UNIV\<close>
 
 definition \<G>_F_L :: \<open>('f \<times> 'l) \<Rightarrow> 'g set\<close> where \<open>\<G>_F_L CL = \<G>_F (fst CL)\<close>
 
 definition \<G>_Inf_L :: \<open>('f \<times> 'l) inference \<Rightarrow> 'g inference set option\<close> where \<open>\<G>_Inf_L \<iota>\<^sub>F\<^sub>L = \<G>_Inf (to_F \<iota>\<^sub>F\<^sub>L)\<close>
 
 (* lem:labeled-grounding-function *)
 sublocale labeled_standard_lifting: standard_lifting
   where
     Bot_F = Bot_FL and
     Inf_F = Inf_FL and
     \<G>_F = \<G>_F_L and
     \<G>_Inf = \<G>_Inf_L
 proof
   show "Bot_FL \<noteq> {}"
     unfolding Bot_FL_def using Bot_F_not_empty by simp
 next
   show "B\<in>Bot_FL \<Longrightarrow> \<G>_F_L B \<noteq> {}" for B
     unfolding \<G>_F_L_def Bot_FL_def using Bot_map_not_empty by auto
 next
   show "B\<in>Bot_FL \<Longrightarrow> \<G>_F_L B \<subseteq> Bot_G" for B
     unfolding \<G>_F_L_def Bot_FL_def using Bot_map by force
 next
   fix CL
   show "\<G>_F_L CL \<inter> Bot_G \<noteq> {} \<longrightarrow> CL \<in> Bot_FL"
     unfolding \<G>_F_L_def Bot_FL_def using Bot_cond
     by (metis SigmaE UNIV_I UNIV_Times_UNIV mem_Sigma_iff prod.sel(1))
 next
   fix \<iota>
   assume
     i_in: \<open>\<iota> \<in> Inf_FL\<close> and
     ground_not_none: \<open>\<G>_Inf_L \<iota> \<noteq> None\<close>
   then show "the (\<G>_Inf_L \<iota>) \<subseteq> Red_Inf_G (\<G>_F_L (concl_of \<iota>))"
     unfolding \<G>_Inf_L_def \<G>_F_L_def to_F_def using inf_map Inf_FL_to_Inf_F by fastforce
 qed
 
 abbreviation Labeled_Empty_Order :: \<open> ('f \<times> 'l) \<Rightarrow> ('f \<times> 'l) \<Rightarrow> bool\<close> where
   "Labeled_Empty_Order C1 C2 \<equiv> False"
 
 sublocale labeled_lifting_w_empty_ord_family :
   lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G entails_G Inf_G Red_Inf_G Red_F_G
     \<G>_F_L \<G>_Inf_L "\<lambda>g. Labeled_Empty_Order"
 proof
   show "po_on Labeled_Empty_Order UNIV"
     unfolding po_on_def by (simp add: transp_onI wfp_on_imp_irreflp_on)
   show "wfp_on Labeled_Empty_Order UNIV"
     unfolding wfp_on_def by simp
 qed
 
 notation "labeled_standard_lifting.entails_\<G>" (infix "\<Turnstile>\<G>L" 50)
 
 (* lem:labeled-consequence *)
 lemma labeled_entailment_lifting: "NL1 \<Turnstile>\<G>L NL2 \<longleftrightarrow> fst ` NL1 \<Turnstile>\<G> fst ` NL2"
   unfolding labeled_standard_lifting.entails_\<G>_def \<G>_F_L_def entails_\<G>_def by auto
 
 lemma (in-) subset_fst: "A \<subseteq> fst ` AB \<Longrightarrow> \<forall>x \<in> A. \<exists>y. (x,y) \<in> AB" by fastforce
 
 lemma red_inf_impl: "\<iota> \<in> labeled_lifting_w_empty_ord_family.Red_Inf_\<G> NL \<Longrightarrow> to_F \<iota> \<in> Red_Inf_\<G> (fst ` NL)"
   unfolding labeled_lifting_w_empty_ord_family.Red_Inf_\<G>_def Red_Inf_\<G>_def \<G>_Inf_L_def \<G>_F_L_def to_F_def
   using Inf_FL_to_Inf_F by auto
 
 (* lem:labeled-saturation *)
 lemma labeled_saturation_lifting:
   "labeled_lifting_w_empty_ord_family.lifted_calculus_with_red_crit.saturated NL \<Longrightarrow>
     empty_order_lifting.lifted_calculus_with_red_crit.saturated (fst ` NL)"
   unfolding labeled_lifting_w_empty_ord_family.lifted_calculus_with_red_crit.saturated_def
     empty_order_lifting.lifted_calculus_with_red_crit.saturated_def
     labeled_standard_lifting.Non_ground.Inf_from_def Non_ground.Inf_from_def
 proof clarify
   fix \<iota>
   assume
     subs_Red_Inf: "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> labeled_lifting_w_empty_ord_family.Red_Inf_\<G> NL" and
     i_in: "\<iota> \<in> Inf_F" and
     i_prems: "set (prems_of \<iota>) \<subseteq> fst ` NL"
   define Lli where "Lli i \<equiv> (SOME x. ((prems_of \<iota>)!i,x) \<in> NL)" for i
   have [simp]:"((prems_of \<iota>)!i,Lli i) \<in> NL" if "i < length (prems_of \<iota>)" for i
     using that subset_fst[OF i_prems] unfolding Lli_def by (meson nth_mem someI_ex)
   define Ll where "Ll \<equiv> map Lli [0..<length (prems_of \<iota>)]"
   have Ll_length: "length Ll = length (prems_of \<iota>)" unfolding Ll_def by auto
   have subs_NL: "set (zip (prems_of \<iota>) Ll) \<subseteq> NL" unfolding Ll_def by (auto simp:in_set_zip)
   obtain L0 where L0: "Infer (zip (prems_of \<iota>) Ll) (concl_of \<iota>, L0) \<in> Inf_FL"
     using Inf_F_to_Inf_FL[OF i_in Ll_length] ..
   define \<iota>_FL where "\<iota>_FL = Infer (zip (prems_of \<iota>) Ll) (concl_of \<iota>, L0)"
   then have "set (prems_of \<iota>_FL) \<subseteq> NL" using subs_NL by simp
   then have "\<iota>_FL \<in> {\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL}" unfolding \<iota>_FL_def using L0 by blast
   then have "\<iota>_FL \<in> labeled_lifting_w_empty_ord_family.Red_Inf_\<G> NL" using subs_Red_Inf by fast
   moreover have "\<iota> = to_F \<iota>_FL" unfolding to_F_def \<iota>_FL_def using Ll_length by (cases \<iota>) auto
   ultimately show "\<iota> \<in> Red_Inf_\<G> (fst ` NL)" by (auto intro:red_inf_impl)
 qed
 
 (* lem:labeled-static-ref-compl *)
 lemma stat_ref_comp_to_labeled_sta_ref_comp: "static_refutational_complete_calculus Bot_F Inf_F (\<Turnstile>\<G>) Red_Inf_\<G> Red_F_\<G> \<Longrightarrow>
   static_refutational_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<G>L)
     labeled_lifting_w_empty_ord_family.Red_Inf_\<G> labeled_lifting_w_empty_ord_family.Red_F_\<G>"
   unfolding static_refutational_complete_calculus_def
 proof (rule conjI impI; clarify)
   interpret calculus_with_red_crit Bot_FL Inf_FL labeled_standard_lifting.entails_\<G>
     labeled_lifting_w_empty_ord_family.Red_Inf_\<G> labeled_lifting_w_empty_ord_family.Red_F_\<G>
     by (simp add:
       labeled_lifting_w_empty_ord_family.lifted_calculus_with_red_crit.calculus_with_red_crit_axioms)
   show "calculus_with_red_crit Bot_FL Inf_FL (\<Turnstile>\<G>L) labeled_lifting_w_empty_ord_family.Red_Inf_\<G>
     labeled_lifting_w_empty_ord_family.Red_F_\<G>"
     by standard
 next
   assume
     calc: "calculus_with_red_crit Bot_F Inf_F (\<Turnstile>\<G>) Red_Inf_\<G> Red_F_\<G>" and
     static: "static_refutational_complete_calculus_axioms Bot_F Inf_F (\<Turnstile>\<G>) Red_Inf_\<G>"
   show "static_refutational_complete_calculus_axioms Bot_FL Inf_FL (\<Turnstile>\<G>L)
     labeled_lifting_w_empty_ord_family.Red_Inf_\<G>"
     unfolding static_refutational_complete_calculus_axioms_def
   proof (intro conjI impI allI)
     fix Bl :: \<open>'f \<times> 'l\<close> and Nl :: \<open>('f \<times> 'l) set\<close>
     assume
       Bl_in: \<open>Bl \<in> Bot_FL\<close> and
       Nl_sat: \<open>labeled_lifting_w_empty_ord_family.lifted_calculus_with_red_crit.saturated Nl\<close> and
       Nl_entails_Bl: \<open>Nl \<Turnstile>\<G>L {Bl}\<close>
     have static_axioms: "B \<in> Bot_F \<longrightarrow> empty_order_lifting.lifted_calculus_with_red_crit.saturated N \<longrightarrow>
       N \<Turnstile>\<G> {B} \<longrightarrow> (\<exists>B'\<in>Bot_F. B' \<in> N)" for B N
       using static[unfolded static_refutational_complete_calculus_axioms_def] by fast
     define B where "B = fst Bl"
     have B_in: "B \<in> Bot_F" using Bl_in Bot_FL_def B_def SigmaE by force
     define N where "N = fst ` Nl"
     have N_sat: "empty_order_lifting.lifted_calculus_with_red_crit.saturated N"
       using N_def Nl_sat labeled_saturation_lifting by blast
     have N_entails_B: "N \<Turnstile>\<G> {B}"
       using Nl_entails_Bl unfolding labeled_entailment_lifting N_def B_def by force
     have "\<exists>B' \<in> Bot_F. B' \<in> N" using B_in N_sat N_entails_B static_axioms[of B N] by blast
     then obtain B' where in_Bot: "B' \<in> Bot_F" and in_N: "B' \<in> N" by force
     then have "B' \<in> fst ` Bot_FL" unfolding Bot_FL_def by fastforce
     obtain Bl' where in_Nl: "Bl' \<in> Nl" and fst_Bl': "fst Bl' = B'"
       using in_N unfolding N_def by blast
     have "Bl' \<in> Bot_FL" unfolding Bot_FL_def using fst_Bl' in_Bot vimage_fst by fastforce
     then show \<open>\<exists>Bl'\<in>Bot_FL. Bl' \<in> Nl\<close> using in_Nl by blast
   qed
 qed
 
 end
 
 subsection \<open>Labeled Lifting with a Family of Redundancy Criteria\<close>
 
 locale labeled_lifting_with_red_crit_family = no_labels: standard_lifting_with_red_crit_family Inf_F
   Bot_G Inf_G Q entails_q Red_Inf_q Red_F_q Bot_F \<G>_F_q \<G>_Inf_q "\<lambda>g. Empty_Order"
   for
     Bot_F :: "'f set" and
     Inf_F :: "'f inference set" and
     Bot_G :: "'g set" and
     Q :: "'q set" and
     entails_q :: "'q \<Rightarrow> 'g set \<Rightarrow> 'g set \<Rightarrow> bool"  and
     Inf_G :: "'g inference set" and
     Red_Inf_q :: "'q \<Rightarrow> 'g set \<Rightarrow> 'g inference set" and
     Red_F_q :: "'q \<Rightarrow> 'g set \<Rightarrow> 'g set" and
     \<G>_F_q :: "'q \<Rightarrow> 'f \<Rightarrow> 'g set" and
     \<G>_Inf_q :: "'q \<Rightarrow> 'f inference \<Rightarrow> 'g inference set option"
   + fixes
-    l :: "'l itself" and
     Inf_FL :: \<open>('f \<times> 'l) inference set\<close>
   assumes
     Inf_F_to_Inf_FL: \<open>\<iota>\<^sub>F \<in> Inf_F \<Longrightarrow> length (Ll :: 'l list) = length (prems_of \<iota>\<^sub>F) \<Longrightarrow> \<exists>L0. Infer (zip (prems_of \<iota>\<^sub>F) Ll) (concl_of \<iota>\<^sub>F, L0) \<in> Inf_FL\<close> and
     Inf_FL_to_Inf_F: \<open>\<iota>\<^sub>F\<^sub>L \<in> Inf_FL \<Longrightarrow> Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L)) \<in> Inf_F\<close>
 begin
 
 definition to_F :: \<open>('f \<times> 'l) inference \<Rightarrow> 'f inference\<close> where
   \<open>to_F \<iota>\<^sub>F\<^sub>L = Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L))\<close>
 
 definition Bot_FL :: \<open>('f \<times> 'l) set\<close> where \<open>Bot_FL = Bot_F \<times> UNIV\<close>
 
 definition \<G>_F_L_q :: \<open>'q \<Rightarrow> ('f \<times> 'l) \<Rightarrow> 'g set\<close> where \<open>\<G>_F_L_q q CL = \<G>_F_q q (fst CL)\<close>
 
 definition \<G>_Inf_L_q :: \<open>'q \<Rightarrow> ('f \<times> 'l) inference \<Rightarrow> 'g inference set option\<close> where
   \<open>\<G>_Inf_L_q q \<iota>\<^sub>F\<^sub>L = \<G>_Inf_q q (to_F \<iota>\<^sub>F\<^sub>L)\<close>
 
 definition \<G>_set_L_q :: "'q \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> 'g set" where
   "\<G>_set_L_q q N \<equiv> \<Union> (\<G>_F_L_q q ` N)"
 
 definition Red_Inf_\<G>_L_q :: "'q \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) inference set" where
   "Red_Inf_\<G>_L_q q N = {\<iota> \<in> Inf_FL. ((\<G>_Inf_L_q q \<iota>) \<noteq> None \<and> the (\<G>_Inf_L_q q \<iota>) \<subseteq> Red_Inf_q q (\<G>_set_L_q q N))
     \<or> ((\<G>_Inf_L_q q \<iota> = None) \<and> \<G>_F_L_q q (concl_of \<iota>) \<subseteq> (\<G>_set_L_q q N \<union> Red_F_q q (\<G>_set_L_q q N)))}"
 
 definition Red_Inf_\<G>_L_Q :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) inference set" where
   "Red_Inf_\<G>_L_Q N = \<Inter> {X N |X. X \<in> (Red_Inf_\<G>_L_q ` Q)}"
 
 definition Labeled_Empty_Order :: \<open> ('f \<times> 'l) \<Rightarrow> ('f \<times> 'l) \<Rightarrow> bool\<close> where
   "Labeled_Empty_Order C1 C2 \<equiv> False"
 
 definition Red_F_\<G>_empty_L_q :: "'q \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "Red_F_\<G>_empty_L_q q N = {C. \<forall>D \<in> \<G>_F_L_q q C. D \<in> Red_F_q q (\<G>_set_L_q q N) \<or>
     (\<exists>E \<in> N. Labeled_Empty_Order E C \<and> D \<in> \<G>_F_L_q q E)}"
 
 definition Red_F_\<G>_empty_L :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "Red_F_\<G>_empty_L N = \<Inter> {X N |X. X \<in> (Red_F_\<G>_empty_L_q ` Q)}"
 
 definition entails_\<G>_L_q :: "'q \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" where
   "entails_\<G>_L_q q N1 N2 \<equiv> entails_q q (\<G>_set_L_q q N1) (\<G>_set_L_q q N2)"
 
 definition entails_\<G>_L_Q :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<Turnstile>\<inter>L" 50) where
   "entails_\<G>_L_Q N1 N2 \<equiv> \<forall>q \<in> Q. entails_\<G>_L_q q N1 N2"
 
 lemma lifting_q:
   assumes "q \<in> Q"
   shows "labeled_lifting_w_wf_ord_family Bot_F Inf_F Bot_G (entails_q q) Inf_G (Red_Inf_q q)
     (Red_F_q q) (\<G>_F_q q) (\<G>_Inf_q q) (\<lambda>g. Empty_Order) Inf_FL"
   using assms no_labels.standard_lifting_family Inf_F_to_Inf_FL Inf_FL_to_Inf_F
   by (simp add: labeled_lifting_w_wf_ord_family_axioms_def labeled_lifting_w_wf_ord_family_def)
 
 lemma lifted_q:
   assumes q_in: "q \<in> Q"
   shows "standard_lifting Bot_FL Inf_FL Bot_G Inf_G (entails_q q) (Red_Inf_q q) (Red_F_q q)
     (\<G>_F_L_q q) (\<G>_Inf_L_q q)"
 proof -
   interpret q_lifting: labeled_lifting_w_wf_ord_family Bot_F Inf_F Bot_G "entails_q q" Inf_G
-    "Red_Inf_q q" "Red_F_q q" "\<G>_F_q q" "\<G>_Inf_q q" "\<lambda>g. Empty_Order" l Inf_FL
+    "Red_Inf_q q" "Red_F_q q" "\<G>_F_q q" "\<G>_Inf_q q" "\<lambda>g. Empty_Order" Inf_FL
     using lifting_q[OF q_in] .
   have "\<G>_F_L_q q = q_lifting.\<G>_F_L"
     unfolding \<G>_F_L_q_def q_lifting.\<G>_F_L_def by simp
   moreover have "\<G>_Inf_L_q q = q_lifting.\<G>_Inf_L"
     unfolding \<G>_Inf_L_q_def q_lifting.\<G>_Inf_L_def to_F_def q_lifting.to_F_def by simp
   moreover have "Bot_FL = q_lifting.Bot_FL"
     unfolding Bot_FL_def q_lifting.Bot_FL_def by simp
   ultimately show "standard_lifting Bot_FL Inf_FL Bot_G Inf_G (entails_q q) (Red_Inf_q q) (Red_F_q q)
     (\<G>_F_L_q q) (\<G>_Inf_L_q q)"
     using q_lifting.labeled_standard_lifting.standard_lifting_axioms by simp
 qed
 
 lemma ord_fam_lifted_q:
   assumes q_in: "q \<in> Q"
   shows "lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G (entails_q q) Inf_G (Red_Inf_q q)
     (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g. Labeled_Empty_Order)"
 proof -
   interpret standard_q_lifting: standard_lifting Bot_FL Inf_FL Bot_G Inf_G "entails_q q"
     "Red_Inf_q q" "Red_F_q q" "\<G>_F_L_q q" "\<G>_Inf_L_q q"
     using lifted_q[OF q_in] .
   have "minimal_element Labeled_Empty_Order UNIV"
     unfolding Labeled_Empty_Order_def
     by (simp add: minimal_element.intro po_on_def transp_onI wfp_on_imp_irreflp_on)
   then show "lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G (entails_q q) Inf_G
     (Red_Inf_q q) (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g. Labeled_Empty_Order)"
     using standard_q_lifting.standard_lifting_axioms
     by (simp add: lifting_with_wf_ordering_family_axioms.intro lifting_with_wf_ordering_family_def)
 qed
 
 lemma all_lifted_red_crit:
   assumes q_in: "q \<in> Q"
   shows "calculus_with_red_crit Bot_FL Inf_FL (entails_\<G>_L_q q) (Red_Inf_\<G>_L_q q)
     (Red_F_\<G>_empty_L_q q)"
 proof -
   interpret ord_q_lifting: lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G "entails_q q" Inf_G
     "Red_Inf_q q" "Red_F_q q" "\<G>_F_L_q q" "\<G>_Inf_L_q q" "\<lambda>g. Labeled_Empty_Order"
     using ord_fam_lifted_q[OF q_in] .
   have "entails_\<G>_L_q q = ord_q_lifting.entails_\<G>"
     unfolding entails_\<G>_L_q_def \<G>_set_L_q_def ord_q_lifting.entails_\<G>_def by simp
   moreover have "Red_Inf_\<G>_L_q q = ord_q_lifting.Red_Inf_\<G>"
     unfolding Red_Inf_\<G>_L_q_def ord_q_lifting.Red_Inf_\<G>_def \<G>_set_L_q_def by simp
   moreover have "Red_F_\<G>_empty_L_q q = ord_q_lifting.Red_F_\<G>"
     unfolding Red_F_\<G>_empty_L_q_def ord_q_lifting.Red_F_\<G>_def \<G>_set_L_q_def by simp
   ultimately show "calculus_with_red_crit Bot_FL Inf_FL (entails_\<G>_L_q q) (Red_Inf_\<G>_L_q q)
     (Red_F_\<G>_empty_L_q q)"
     using ord_q_lifting.lifted_calculus_with_red_crit.calculus_with_red_crit_axioms by argo
 qed
 
 lemma all_lifted_cons_rel:
   assumes q_in: "q \<in> Q"
   shows "consequence_relation Bot_FL (entails_\<G>_L_q q)"
 proof -
   interpret q_red_crit: calculus_with_red_crit Bot_FL Inf_FL "entails_\<G>_L_q q" "Red_Inf_\<G>_L_q q"
     "Red_F_\<G>_empty_L_q q"
     using all_lifted_red_crit[OF q_in] .
   show "consequence_relation Bot_FL (entails_\<G>_L_q q)"
     using q_red_crit.consequence_relation_axioms .
 qed
 
 sublocale labeled_cons_rel_family: consequence_relation_family Bot_FL Q entails_\<G>_L_q
   using all_lifted_cons_rel no_labels.lifted_calc_w_red_crit_family.bot_not_empty
   by (simp add: Bot_FL_def consequence_relation_family_def
       no_labels.lifted_calc_w_red_crit_family.Q_nonempty)
 
 sublocale with_labels: calculus_with_red_crit_family Bot_FL Inf_FL Q entails_\<G>_L_q Red_Inf_\<G>_L_q
   Red_F_\<G>_empty_L_q
   using calculus_with_red_crit_family.intro[OF labeled_cons_rel_family.consequence_relation_family_axioms]
     all_lifted_cons_rel
   by (meson calculus_with_red_crit_family_axioms.intro
       labeled_lifting_with_red_crit_family.all_lifted_red_crit
       labeled_lifting_with_red_crit_family_axioms no_labels.lifted_calc_w_red_crit_family.Q_nonempty)
 
 notation no_labels.entails_\<G>_Q (infix "\<Turnstile>\<inter>" 50)
 
 (* lem:labeled-consequence-intersection *)
 lemma labeled_entailment_lifting: "NL1 \<Turnstile>\<inter>L NL2 \<longleftrightarrow> fst ` NL1 \<Turnstile>\<inter> fst ` NL2"
   unfolding no_labels.entails_\<G>_Q_def no_labels.entails_\<G>_q_def no_labels.\<G>_set_q_def
     entails_\<G>_L_Q_def entails_\<G>_L_q_def \<G>_set_L_q_def \<G>_F_L_q_def
   by force
 
 lemma subset_fst: "A \<subseteq> fst ` AB \<Longrightarrow> \<forall>x \<in> A. \<exists>y. (x,y) \<in> AB" by fastforce
 
 lemma red_inf_impl: "\<iota> \<in> with_labels.Red_Inf_Q NL \<Longrightarrow>
   to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` NL)"
   unfolding no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q_def with_labels.Red_Inf_Q_def
 proof clarify
   fix X Xa q
   assume
     q_in: "q \<in> Q" and
     i_in_inter: "\<iota> \<in> \<Inter> {X NL |X. X \<in> Red_Inf_\<G>_L_q ` Q}"
   have i_in_q: "\<iota> \<in> Red_Inf_\<G>_L_q q NL" using q_in i_in_inter image_eqI by blast
   then have i_in: "\<iota> \<in> Inf_FL" unfolding Red_Inf_\<G>_L_q_def by blast
   have to_F_in: "to_F \<iota> \<in> Inf_F" unfolding to_F_def using Inf_FL_to_Inf_F[OF i_in] .
   have rephrase1: "(\<Union>CL\<in>NL. \<G>_F_q q (fst CL)) = (\<Union> (\<G>_F_q q ` fst ` NL))" by blast
   have rephrase2: "fst (concl_of \<iota>) = concl_of (to_F \<iota>)"
     unfolding concl_of_def to_F_def by simp
   have subs_red: "((\<G>_Inf_L_q q \<iota>) \<noteq> None \<and> the (\<G>_Inf_L_q q \<iota>) \<subseteq> Red_Inf_q q (\<G>_set_L_q q NL))
     \<or> ((\<G>_Inf_L_q q \<iota> = None) \<and> \<G>_F_L_q q (concl_of \<iota>) \<subseteq> (\<G>_set_L_q q NL \<union> Red_F_q q (\<G>_set_L_q q NL)))"
     using i_in_q unfolding Red_Inf_\<G>_L_q_def by blast
   then have to_F_subs_red: "(\<G>_Inf_q q (to_F \<iota>) \<noteq> None \<and>
       the (\<G>_Inf_q q (to_F \<iota>)) \<subseteq> Red_Inf_q q (no_labels.\<G>_set_q q (fst ` NL)))
     \<or> (\<G>_Inf_q q (to_F \<iota>) = None \<and>
       \<G>_F_q q (concl_of (to_F \<iota>)) \<subseteq> (no_labels.\<G>_set_q q (fst ` NL) \<union> Red_F_q q (no_labels.\<G>_set_q q (fst ` NL))))"
     unfolding \<G>_Inf_L_q_def \<G>_set_L_q_def no_labels.\<G>_set_q_def \<G>_F_L_q_def
     using rephrase1 rephrase2 by metis
   then show "to_F \<iota> \<in> no_labels.Red_Inf_\<G>_q q (fst ` NL)"
     using to_F_in unfolding no_labels.Red_Inf_\<G>_q_def by simp
 qed
 
 (* lem:labeled-saturation-intersection *)
 lemma labeled_family_saturation_lifting: "with_labels.inter_red_crit_calculus.saturated NL \<Longrightarrow>
   no_labels.lifted_calc_w_red_crit_family.inter_red_crit_calculus.saturated (fst ` NL)"
   unfolding with_labels.inter_red_crit_calculus.saturated_def
     no_labels.lifted_calc_w_red_crit_family.inter_red_crit_calculus.saturated_def
     with_labels.Inf_from_def no_labels.Non_ground.Inf_from_def
 proof clarify
   fix \<iota>F
   assume
     labeled_sat: "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> with_labels.Red_Inf_Q NL" and
     iF_in: "\<iota>F \<in> Inf_F" and
     iF_prems: "set (prems_of \<iota>F) \<subseteq> fst ` NL"
   define Lli where "Lli i \<equiv> (SOME x. ((prems_of \<iota>F)!i,x) \<in> NL)" for i
   have [simp]:"((prems_of \<iota>F)!i,Lli i) \<in> NL" if "i < length (prems_of \<iota>F)" for i
     using that subset_fst[OF iF_prems] nth_mem someI_ex unfolding Lli_def
     by metis
   define Ll where "Ll \<equiv> map Lli [0..<length (prems_of \<iota>F)]"
   have Ll_length: "length Ll = length (prems_of \<iota>F)" unfolding Ll_def by auto
   have subs_NL: "set (zip (prems_of \<iota>F) Ll) \<subseteq> NL" unfolding Ll_def by (auto simp:in_set_zip)
   obtain L0 where L0: "Infer (zip (prems_of \<iota>F) Ll) (concl_of \<iota>F, L0) \<in> Inf_FL"
     using Inf_F_to_Inf_FL[OF iF_in Ll_length] ..
   define \<iota>FL where "\<iota>FL = Infer (zip (prems_of \<iota>F) Ll) (concl_of \<iota>F, L0)"
   then have "set (prems_of \<iota>FL) \<subseteq> NL" using subs_NL by simp
   then have "\<iota>FL \<in> {\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL}" unfolding \<iota>FL_def using L0 by blast
   then have "\<iota>FL \<in> with_labels.Red_Inf_Q NL" using labeled_sat by fast
   moreover have "\<iota>F = to_F \<iota>FL" unfolding to_F_def \<iota>FL_def using Ll_length by (cases \<iota>F) auto
   ultimately show "\<iota>F \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` NL)"
     by (auto intro:red_inf_impl)
 qed
 
 (* thm:labeled-static-ref-compl-intersection *)
 theorem labeled_static_ref: "static_refutational_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>)
   no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q
   no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q
   \<Longrightarrow> static_refutational_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<inter>L) with_labels.Red_Inf_Q with_labels.Red_F_Q"
   unfolding static_refutational_complete_calculus_def
 proof (rule conjI impI; clarify)
   show "calculus_with_red_crit Bot_FL Inf_FL (\<Turnstile>\<inter>L) with_labels.Red_Inf_Q with_labels.Red_F_Q"
     using with_labels.inter_red_crit_calculus.calculus_with_red_crit_axioms
     unfolding labeled_cons_rel_family.entails_Q_def entails_\<G>_L_Q_def .
 next
   assume
     calc: "calculus_with_red_crit Bot_F Inf_F (\<Turnstile>\<inter>)
       no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q
       no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q" and
     static: "static_refutational_complete_calculus_axioms Bot_F Inf_F (\<Turnstile>\<inter>)
       no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q"
   show "static_refutational_complete_calculus_axioms Bot_FL Inf_FL (\<Turnstile>\<inter>L) with_labels.Red_Inf_Q"
     unfolding static_refutational_complete_calculus_axioms_def
   proof (intro conjI impI allI)
     fix Bl :: \<open>'f \<times> 'l\<close> and Nl :: \<open>('f \<times> 'l) set\<close>
     assume
       Bl_in: \<open>Bl \<in> Bot_FL\<close> and
       Nl_sat: \<open>with_labels.inter_red_crit_calculus.saturated Nl\<close> and
       Nl_entails_Bl: \<open>Nl \<Turnstile>\<inter>L {Bl}\<close>
     have static_axioms: "B \<in> Bot_F \<longrightarrow>
       no_labels.lifted_calc_w_red_crit_family.inter_red_crit_calculus.saturated N \<longrightarrow>
       N \<Turnstile>\<inter> {B} \<longrightarrow> (\<exists>B'\<in>Bot_F. B' \<in> N)" for B N
       using static[unfolded static_refutational_complete_calculus_axioms_def] by fast
     define B where "B = fst Bl"
     have B_in: "B \<in> Bot_F" using Bl_in Bot_FL_def B_def SigmaE by force
     define N where "N = fst ` Nl"
     have N_sat: "no_labels.lifted_calc_w_red_crit_family.inter_red_crit_calculus.saturated N"
       using N_def Nl_sat labeled_family_saturation_lifting by blast
     have N_entails_B: "N \<Turnstile>\<inter> {B}"
       using Nl_entails_Bl unfolding labeled_entailment_lifting N_def B_def by force
     have "\<exists>B' \<in> Bot_F. B' \<in> N" using B_in N_sat N_entails_B static_axioms[of B N] by blast
     then obtain B' where in_Bot: "B' \<in> Bot_F" and in_N: "B' \<in> N" by force
     then have "B' \<in> fst ` Bot_FL" unfolding Bot_FL_def by fastforce
     obtain Bl' where in_Nl: "Bl' \<in> Nl" and fst_Bl': "fst Bl' = B'"
       using in_N unfolding N_def by blast
     have "Bl' \<in> Bot_FL" unfolding Bot_FL_def using fst_Bl' in_Bot vimage_fst by fastforce
     then show \<open>\<exists>Bl'\<in>Bot_FL. Bl' \<in> Nl\<close> using in_Nl by blast
   qed
 qed
 
 end
 
 end
diff --git a/thys/Saturation_Framework/Prover_Architectures.thy b/thys/Saturation_Framework/Prover_Architectures.thy
--- a/thys/Saturation_Framework/Prover_Architectures.thy
+++ b/thys/Saturation_Framework/Prover_Architectures.thy
@@ -1,1307 +1,1304 @@
 (*  Title:       Prover Architectures of the Saturation Framework
  *  Author:      Sophie Tourret <stourret at mpi-inf.mpg.de>, 2019-2020 *)
 
 section \<open>Prover Architectures\<close>
 
 text \<open>This section covers all the results presented in the section 4 of the report.
   This is where abstract architectures of provers are defined and proven
   dynamically refutationally complete.\<close>
 
 theory Prover_Architectures
   imports Labeled_Lifting_to_Non_Ground_Calculi
 begin
 
 subsection \<open>Basis of the Prover Architectures\<close>
 
 locale Prover_Architecture_Basis = labeled_lifting_with_red_crit_family Bot_F Inf_F Bot_G Q entails_q Inf_G
-  Red_Inf_q Red_F_q \<G>_F_q \<G>_Inf_q l Inf_FL
+  Red_Inf_q Red_F_q \<G>_F_q \<G>_Inf_q Inf_FL
   for
     Bot_F :: "'f set"
     and Inf_F :: "'f inference set"
     and Bot_G :: "'g set"
     and Q :: "'q set"
     and entails_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set \<Rightarrow> bool)"
     and Inf_G :: \<open>'g inference set\<close>
     and Red_Inf_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g inference set)"
     and Red_F_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set)"
     and \<G>_F_q :: "'q \<Rightarrow> 'f \<Rightarrow> 'g set"
     and \<G>_Inf_q :: "'q \<Rightarrow> 'f inference \<Rightarrow> 'g inference set option"
-    and l :: "'l itself"
     and Inf_FL :: \<open>('f \<times> 'l) inference set\<close>
   + fixes
     Equiv_F :: "('f \<times> 'f) set" and
     Prec_F :: "'f \<Rightarrow> 'f \<Rightarrow> bool" (infix "\<cdot>\<succ>" 50) and
     Prec_l :: "'l \<Rightarrow> 'l \<Rightarrow> bool" (infix "\<sqsubset>l" 50)
   assumes
     equiv_F_is_equiv_rel: "equiv UNIV Equiv_F" and
     wf_prec_F: "minimal_element (Prec_F) UNIV" and
     wf_prec_l: "minimal_element (Prec_l) UNIV" and
     compat_equiv_prec: "(C1, D1) \<in> equiv_F \<Longrightarrow> (C2,D2) \<in> equiv_F \<Longrightarrow> C1 \<cdot>\<succ> C2 \<Longrightarrow> D1 \<cdot>\<succ> D2" and
     equiv_F_grounding: "q \<in> Q \<Longrightarrow> (C1, C2) \<in> equiv_F \<Longrightarrow> \<G>_F_q q C1 = \<G>_F_q q C2" and
     prec_F_grounding: "q \<in> Q \<Longrightarrow> C1 \<cdot>\<succ> C2 \<Longrightarrow> \<G>_F_q q C1 \<subseteq> \<G>_F_q q C2" and
     static_ref_comp: "static_refutational_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>)
       no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q
       no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q"
 begin
 
 definition equiv_F_fun :: "'f \<Rightarrow> 'f \<Rightarrow> bool" (infix "\<doteq>" 50) where
   "equiv_F_fun C D \<equiv> (C, D) \<in> Equiv_F"
 
 definition Prec_eq_F :: "'f \<Rightarrow> 'f \<Rightarrow> bool" (infix "\<cdot>\<succeq>" 50) where
   "Prec_eq_F C D \<equiv> ((C, D) \<in> Equiv_F \<or> C \<cdot>\<succ> D)"
 
 definition Prec_FL :: "('f \<times> 'l) \<Rightarrow> ('f \<times> 'l) \<Rightarrow> bool" (infix "\<sqsubset>" 50) where
   "Prec_FL Cl1 Cl2 \<equiv> (fst Cl1 \<cdot>\<succ> fst Cl2) \<or> (fst Cl1 \<doteq> fst Cl2 \<and> snd Cl1 \<sqsubset>l snd Cl2)"
 
 lemma wf_prec_FL: "minimal_element (\<sqsubset>) UNIV"
 proof
   show "po_on (\<sqsubset>) UNIV" unfolding po_on_def
   proof
     show "irreflp_on (\<sqsubset>) UNIV" unfolding irreflp_on_def Prec_FL_def
     proof
       fix a
       assume a_in: "a \<in> (UNIV::('f \<times> 'l) set)"
       have "\<not> (fst a \<cdot>\<succ> fst a)" using wf_prec_F minimal_element.min_elt_ex by force
       moreover have "\<not> (snd a \<sqsubset>l snd a)" using wf_prec_l minimal_element.min_elt_ex by force
       ultimately show "\<not> (fst a \<cdot>\<succ> fst a \<or> fst a \<doteq> fst a \<and> snd a \<sqsubset>l snd a)" by blast
     qed
   next
     show "transp_on (\<sqsubset>) UNIV" unfolding transp_on_def Prec_FL_def
     proof (simp, intro allI impI)
       fix a1 b1 a2 b2 a3 b3
       assume trans_hyp:"(a1 \<cdot>\<succ> a2 \<or> a1 \<doteq> a2 \<and> b1 \<sqsubset>l b2) \<and> (a2 \<cdot>\<succ> a3 \<or> a2 \<doteq> a3 \<and> b2 \<sqsubset>l b3)"
       have "a1 \<cdot>\<succ> a2 \<Longrightarrow> a2 \<cdot>\<succ> a3 \<Longrightarrow> a1 \<cdot>\<succ> a3" using wf_prec_F compat_equiv_prec by blast
       moreover have "a1 \<cdot>\<succ> a2 \<Longrightarrow> a2 \<doteq> a3 \<Longrightarrow> a1 \<cdot>\<succ> a3" using wf_prec_F compat_equiv_prec by blast
       moreover have "a1 \<doteq> a2 \<Longrightarrow> a2 \<cdot>\<succ> a3 \<Longrightarrow> a1 \<cdot>\<succ> a3" using wf_prec_F compat_equiv_prec by blast
       moreover have "b1 \<sqsubset>l b2 \<Longrightarrow> b2 \<sqsubset>l b3 \<Longrightarrow> b1 \<sqsubset>l b3"
         using wf_prec_l unfolding minimal_element_def po_on_def transp_on_def by (meson UNIV_I)
       moreover have "a1 \<doteq> a2 \<Longrightarrow> a2 \<doteq> a3 \<Longrightarrow> a1 \<doteq> a3"
         using equiv_F_is_equiv_rel equiv_class_eq unfolding equiv_F_fun_def by fastforce
       ultimately show "(a1 \<cdot>\<succ> a3 \<or> a1 \<doteq> a3 \<and> b1 \<sqsubset>l b3)" using trans_hyp by blast
     qed
   qed
 next
   show "wfp_on (\<sqsubset>) UNIV" unfolding wfp_on_def
   proof
     assume contra: "\<exists>f. \<forall>i. f i \<in> UNIV \<and> f (Suc i) \<sqsubset> f i"
     then obtain f where f_in: "\<forall>i. f i \<in> UNIV" and f_suc: "\<forall>i. f (Suc i) \<sqsubset> f i" by blast
     define f_F where "f_F = (\<lambda>i. fst (f i))"
     define f_L where "f_L = (\<lambda>i. snd (f i))"
     have uni_F: "\<forall>i. f_F i \<in> UNIV" using f_in by simp
     have uni_L: "\<forall>i. f_L i \<in> UNIV" using f_in by simp
     have decomp: "\<forall>i. f_F (Suc i) \<cdot>\<succ> f_F i \<or> f_L (Suc i) \<sqsubset>l f_L i"
       using f_suc unfolding Prec_FL_def f_F_def f_L_def by blast
     define I_F where "I_F = { i |i. f_F (Suc i) \<cdot>\<succ> f_F i}"
     define I_L where "I_L = { i |i. f_L (Suc i) \<sqsubset>l f_L i}"
     have "I_F \<union> I_L = UNIV" using decomp unfolding I_F_def I_L_def by blast
     then have "finite I_F \<Longrightarrow> \<not> finite I_L" by (metis finite_UnI infinite_UNIV_nat)
     moreover have "infinite I_F \<Longrightarrow> \<exists>f. \<forall>i. f i \<in> UNIV \<and> f (Suc i) \<cdot>\<succ> f i"
       using uni_F unfolding I_F_def by (meson compat_equiv_prec iso_tuple_UNIV_I not_finite_existsD)
     moreover have "infinite I_L \<Longrightarrow> \<exists>f. \<forall>i. f i \<in> UNIV \<and> f (Suc i) \<sqsubset>l f i"
       using uni_L unfolding I_L_def
       by (metis UNIV_I compat_equiv_prec decomp minimal_element_def wf_prec_F wfp_on_def)
     ultimately show False using wf_prec_F wf_prec_l by (metis minimal_element_def wfp_on_def)
   qed
 qed
 
 lemma labeled_static_ref_comp:
   "static_refutational_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<inter>L) with_labels.Red_Inf_Q with_labels.Red_F_Q"
   using labeled_static_ref[OF static_ref_comp] .
 
 lemma standard_labeled_lifting_family:
   assumes q_in: "q \<in> Q"
   shows "lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G
     (entails_q q) Inf_G (Red_Inf_q q) (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g. Prec_FL)"
 proof -
   have "lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G (entails_q q) Inf_G
     (Red_Inf_q q) (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g. Labeled_Empty_Order)"
     using ord_fam_lifted_q[OF q_in] .
   then have "standard_lifting Bot_FL Inf_FL Bot_G Inf_G (entails_q q) (Red_Inf_q q) (Red_F_q q)
     (\<G>_F_L_q q) (\<G>_Inf_L_q q)"
     using lifted_q[OF q_in] by blast
   then show "lifting_with_wf_ordering_family Bot_FL Inf_FL Bot_G (entails_q q) Inf_G (Red_Inf_q q)
     (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g. Prec_FL)"
     using wf_prec_FL
     by (simp add: lifting_with_wf_ordering_family.intro lifting_with_wf_ordering_family_axioms.intro)
 qed
 
 sublocale labeled_ord_red_crit_fam: standard_lifting_with_red_crit_family Inf_FL Bot_G Inf_G Q
   entails_q Red_Inf_q Red_F_q
   Bot_FL \<G>_F_L_q \<G>_Inf_L_q "\<lambda>g. Prec_FL"
   using standard_labeled_lifting_family
     no_labels.Ground_family.calculus_with_red_crit_family_axioms
   by (simp add: standard_lifting_with_red_crit_family.intro
     standard_lifting_with_red_crit_family_axioms.intro)
 
 lemma entail_equiv:
   "labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q N1 N2 = (N1 \<Turnstile>\<inter>L N2)"
   unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q_def
     entails_\<G>_L_Q_def entails_\<G>_L_q_def labeled_ord_red_crit_fam.entails_\<G>_q_def
      labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_set_L_q_def
   by simp
 
 lemma entail_equiv2: "labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q = (\<Turnstile>\<inter>L)"
   using entail_equiv by auto
 
 lemma red_inf_equiv: "labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q N =
   with_labels.Red_Inf_Q N"
   unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q_def
     with_labels.Red_Inf_Q_def labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def Red_Inf_\<G>_L_q_def
     labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_set_L_q_def
   by simp
 
 lemma red_inf_equiv2: "labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q =
   with_labels.Red_Inf_Q"
   using red_inf_equiv by auto
 
 lemma empty_red_f_equiv: "labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q N =
   with_labels.Red_F_Q N"
   unfolding labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q_def
     with_labels.Red_F_Q_def labeled_ord_red_crit_fam.Red_F_\<G>_empty_q_def Red_F_\<G>_empty_L_q_def
     labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_set_L_q_def Labeled_Empty_Order_def
   by simp
 
 lemma empty_red_f_equiv2: "labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_F_Q =
   with_labels.Red_F_Q"
   using empty_red_f_equiv by auto
 
 lemma labeled_ordered_static_ref_comp:
   "static_refutational_complete_calculus Bot_FL Inf_FL
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q"
   using labeled_ord_red_crit_fam.static_empty_ord_inter_equiv_static_inter empty_red_f_equiv2
     red_inf_equiv2 entail_equiv2 labeled_static_ref_comp
   by argo
 
 interpretation stat_ref_calc: static_refutational_complete_calculus Bot_FL Inf_FL
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q
   by (rule labeled_ordered_static_ref_comp)
 
 lemma labeled_ordered_dynamic_ref_comp:
   "dynamic_refutational_complete_calculus Bot_FL Inf_FL
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.entails_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q
   labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q"
   by (rule stat_ref_calc.dynamic_refutational_complete_calculus_axioms)
 
 (* lem:redundant-labeled-inferences *)
 lemma labeled_red_inf_eq_red_inf: "\<iota> \<in> Inf_FL \<Longrightarrow>
   \<iota> \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q N \<equiv>
   to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N)" for \<iota>
 proof -
   fix \<iota>
   assume i_in: "\<iota> \<in> Inf_FL"
   have "\<iota> \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q N \<Longrightarrow>
     to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N)"
   proof -
     assume i_in2: "\<iota> \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q N"
     then have "X \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q ` Q \<Longrightarrow> \<iota> \<in> X N" for X
       unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q_def by blast
     obtain X0 where "X0 \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q ` Q"
       using with_labels.Q_nonempty by blast
     then obtain q0 where x0_is: "X0 N = labeled_ord_red_crit_fam.Red_Inf_\<G>_q q0 N" by blast
     then obtain Y0 where y0_is: "Y0 (fst ` N) = to_F ` (X0 N)" by auto
     have "Y0 (fst ` N) = no_labels.Red_Inf_\<G>_q q0 (fst ` N)"
       unfolding  y0_is
     proof
       show "to_F ` X0 N \<subseteq> no_labels.Red_Inf_\<G>_q q0 (fst ` N)"
       proof
         fix \<iota>0
         assume i0_in: "\<iota>0 \<in> to_F ` X0 N"
         then have i0_in2: "\<iota>0 \<in> to_F ` (labeled_ord_red_crit_fam.Red_Inf_\<G>_q q0 N)"
           using x0_is by argo
         then obtain \<iota>0_FL where i0_FL_in: "\<iota>0_FL \<in> Inf_FL" and i0_to_i0_FL: "\<iota>0 = to_F \<iota>0_FL" and
           subs1: "((\<G>_Inf_L_q q0 \<iota>0_FL) \<noteq> None \<and>
             the (\<G>_Inf_L_q q0 \<iota>0_FL) \<subseteq> Red_Inf_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N))
             \<or> ((\<G>_Inf_L_q q0 \<iota>0_FL = None) \<and>
             \<G>_F_L_q q0 (concl_of \<iota>0_FL) \<subseteq> (labeled_ord_red_crit_fam.\<G>_set_q q0 N \<union>
               Red_F_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N)))"
           unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def by blast
         have concl_swap: "fst (concl_of \<iota>0_FL) = concl_of \<iota>0"
           unfolding concl_of_def i0_to_i0_FL to_F_def by simp
         have i0_in3: "\<iota>0 \<in> Inf_F"
           using i0_to_i0_FL Inf_FL_to_Inf_F[OF i0_FL_in] unfolding to_F_def by blast
         {
           assume
             not_none: "\<G>_Inf_q q0 \<iota>0 \<noteq> None" and
             "the (\<G>_Inf_q q0 \<iota>0) \<noteq> {}"
           then obtain \<iota>1 where i1_in: "\<iota>1 \<in> the (\<G>_Inf_q q0 \<iota>0)" by blast
           have "the (\<G>_Inf_q q0 \<iota>0) \<subseteq> Red_Inf_q q0 (no_labels.\<G>_set_q q0 (fst ` N))"
             using subs1 i0_to_i0_FL not_none
             unfolding no_labels.\<G>_set_q_def labeled_ord_red_crit_fam.\<G>_set_q_def
               \<G>_Inf_L_q_def \<G>_F_L_q_def by auto
         }
         moreover {
           assume
             is_none: "\<G>_Inf_q q0 \<iota>0 = None"
           then have "\<G>_F_q q0 (concl_of \<iota>0) \<subseteq> no_labels.\<G>_set_q q0 (fst ` N)
             \<union> Red_F_q q0 (no_labels.\<G>_set_q q0 (fst ` N))"
             using subs1 i0_to_i0_FL concl_swap
             unfolding no_labels.\<G>_set_q_def labeled_ord_red_crit_fam.\<G>_set_q_def
               \<G>_Inf_L_q_def \<G>_F_L_q_def by simp
         }
         ultimately show "\<iota>0 \<in> no_labels.Red_Inf_\<G>_q q0 (fst ` N)"
           unfolding no_labels.Red_Inf_\<G>_q_def using i0_in3 by auto
        qed
      next
        show "no_labels.Red_Inf_\<G>_q q0 (fst ` N) \<subseteq> to_F ` X0 N"
        proof
          fix \<iota>0
          assume i0_in: "\<iota>0 \<in> no_labels.Red_Inf_\<G>_q q0 (fst ` N)"
          then have i0_in2: "\<iota>0 \<in> Inf_F"
            unfolding no_labels.Red_Inf_\<G>_q_def by blast
          obtain \<iota>0_FL where i0_FL_in: "\<iota>0_FL \<in> Inf_FL" and i0_to_i0_FL: "\<iota>0 = to_F \<iota>0_FL"
            using Inf_F_to_Inf_FL[OF i0_in2] unfolding to_F_def
            by (metis Ex_list_of_length fst_conv inference.exhaust_sel inference.inject map_fst_zip)
          have concl_swap: "fst (concl_of \<iota>0_FL) = concl_of \<iota>0"
            unfolding concl_of_def i0_to_i0_FL to_F_def by simp
          have subs1: "((\<G>_Inf_L_q q0 \<iota>0_FL) \<noteq> None \<and>
            the (\<G>_Inf_L_q q0 \<iota>0_FL) \<subseteq> Red_Inf_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N))
            \<or> ((\<G>_Inf_L_q q0 \<iota>0_FL = None) \<and>
            \<G>_F_L_q q0 (concl_of \<iota>0_FL) \<subseteq> (labeled_ord_red_crit_fam.\<G>_set_q q0 N \<union>
              Red_F_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N)))"
            using i0_in i0_to_i0_FL concl_swap
            unfolding no_labels.Red_Inf_\<G>_q_def \<G>_Inf_L_q_def no_labels.\<G>_set_q_def
              labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_F_L_q_def
            by simp
          then have "\<iota>0_FL \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q q0 N"
            using i0_FL_in unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def
            by simp
          then show "\<iota>0 \<in> to_F ` X0 N"
            using x0_is i0_to_i0_FL i0_in2 by blast
        qed
      qed
     then have "Y \<in> no_labels.Red_Inf_\<G>_q ` Q \<Longrightarrow> (to_F \<iota>) \<in> Y (fst ` N)" for Y
       using i_in2 no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q_def
         red_inf_equiv2 red_inf_impl by fastforce
     then show "(to_F \<iota>) \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N)"
       unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q_def
         no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q_def
       by blast
     qed
   moreover have "(to_F \<iota>) \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N) \<Longrightarrow>
     \<iota> \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q N"
   proof -
     assume to_F_in: "to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N)"
     have imp_to_F: "X \<in> no_labels.Red_Inf_\<G>_q ` Q \<Longrightarrow> to_F \<iota> \<in> X (fst ` N)" for X
       using to_F_in unfolding no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q_def
       by blast
     then have to_F_in2: "to_F \<iota> \<in> no_labels.Red_Inf_\<G>_q q (fst ` N)" if "q \<in> Q" for q
       using that by auto
     have "labeled_ord_red_crit_fam.Red_Inf_\<G>_q q N =
       {\<iota>0_FL \<in> Inf_FL. to_F \<iota>0_FL \<in> no_labels.Red_Inf_\<G>_q q (fst ` N)}" for q
     proof
       show "labeled_ord_red_crit_fam.Red_Inf_\<G>_q q N \<subseteq>
         {\<iota>0_FL \<in> Inf_FL. to_F \<iota>0_FL \<in> no_labels.Red_Inf_\<G>_q q (fst ` N)}"
       proof
         fix q0 \<iota>1
         assume
           i1_in: "\<iota>1 \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q q0 N"
         have i1_in2: "\<iota>1 \<in> Inf_FL"
           using i1_in unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def by blast
           then have to_F_i1_in: "to_F \<iota>1 \<in> Inf_F"
           using Inf_FL_to_Inf_F unfolding to_F_def by simp
         have concl_swap: "fst (concl_of \<iota>1) = concl_of (to_F \<iota>1)"
           unfolding concl_of_def to_F_def by simp
         then have i1_to_F_in: "to_F \<iota>1 \<in> no_labels.Red_Inf_\<G>_q q0 (fst ` N)"
           using i1_in to_F_i1_in
           unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def no_labels.Red_Inf_\<G>_q_def
             \<G>_Inf_L_q_def labeled_ord_red_crit_fam.\<G>_set_q_def no_labels.\<G>_set_q_def \<G>_F_L_q_def
             by force
         show "\<iota>1 \<in> {\<iota>0_FL \<in> Inf_FL. to_F \<iota>0_FL \<in> no_labels.Red_Inf_\<G>_q q0 (fst ` N)}"
           using i1_in2 i1_to_F_in by blast
       qed
     next
       show "{\<iota>0_FL \<in> Inf_FL. to_F \<iota>0_FL \<in> no_labels.Red_Inf_\<G>_q q (fst ` N)} \<subseteq>
         labeled_ord_red_crit_fam.Red_Inf_\<G>_q q N"
       proof
         fix q0 \<iota>1
         assume
           i1_in: "\<iota>1 \<in> {\<iota>0_FL \<in> Inf_FL. to_F \<iota>0_FL \<in> no_labels.Red_Inf_\<G>_q q0 (fst ` N)}"
         then have i1_in2: "\<iota>1 \<in> Inf_FL" by blast
         then have to_F_i1_in: "to_F \<iota>1 \<in> Inf_F"
           using Inf_FL_to_Inf_F unfolding to_F_def by simp
         have concl_swap: "fst (concl_of \<iota>1) = concl_of (to_F \<iota>1)"
           unfolding concl_of_def to_F_def by simp
         then have "((\<G>_Inf_L_q q0 \<iota>1) \<noteq> None \<and>
           the (\<G>_Inf_L_q q0 \<iota>1) \<subseteq> Red_Inf_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N))
           \<or> ((\<G>_Inf_L_q q0 \<iota>1 = None) \<and>
           \<G>_F_L_q q0 (concl_of \<iota>1) \<subseteq> (labeled_ord_red_crit_fam.\<G>_set_q q0 N \<union>
             Red_F_q q0 (labeled_ord_red_crit_fam.\<G>_set_q q0 N)))"
           using i1_in unfolding no_labels.Red_Inf_\<G>_q_def \<G>_Inf_L_q_def
             labeled_ord_red_crit_fam.\<G>_set_q_def no_labels.\<G>_set_q_def \<G>_F_L_q_def
           by auto
         then show "\<iota>1 \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q q0 N"
           using i1_in2 unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def
           by blast
       qed
     qed
     then have "\<iota> \<in> labeled_ord_red_crit_fam.Red_Inf_\<G>_q q N" if "q \<in> Q" for q
       using that to_F_in2 i_in
       unfolding labeled_ord_red_crit_fam.Red_Inf_\<G>_q_def
         no_labels.Red_Inf_\<G>_q_def \<G>_Inf_L_q_def labeled_ord_red_crit_fam.\<G>_set_q_def
         no_labels.\<G>_set_q_def \<G>_F_L_q_def
       by auto
     then show "\<iota> \<in> labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q N"
       unfolding labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q_def
       by blast
   qed
   ultimately show "\<iota> \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_Inf_Q N \<equiv>
     to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` N)"
     by argo
 qed
 
 (* lem:redundant-labeled-formulas *)
 lemma red_labeled_clauses: \<open>C \<in> no_labels.Red_F_\<G>_empty (fst ` N) \<or> (\<exists>C' \<in> (fst ` N). C \<cdot>\<succ> C') \<or>
   (\<exists>(C', L') \<in> N. (L' \<sqsubset>l L \<and> C \<cdot>\<succeq> C')) \<Longrightarrow>
   (C, L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N\<close>
 proof -
   assume \<open>C \<in> no_labels.Red_F_\<G>_empty (fst ` N) \<or>
     (\<exists>C' \<in> (fst ` N). C \<cdot>\<succ> C') \<or> (\<exists>(C',L') \<in> N. (L' \<sqsubset>l L \<and> C \<cdot>\<succeq> C'))\<close>
   moreover have i: \<open>C \<in> no_labels.Red_F_\<G>_empty (fst ` N) \<Longrightarrow>
     (C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N\<close>
   proof -
     assume "C \<in> no_labels.Red_F_\<G>_empty (fst ` N)"
     then have "C \<in> no_labels.Red_F_\<G>_empty_q q (fst ` N)" if "q \<in> Q" for q
       unfolding no_labels.Red_F_\<G>_empty_def using that by fast
     then have g_in_red: "\<G>_F_q q C \<subseteq> Red_F_q q (no_labels.\<G>_set_q q (fst ` N))" if "q \<in> Q" for q
       unfolding no_labels.Red_F_\<G>_empty_q_def using that by blast
     have "no_labels.\<G>_set_q q (fst ` N) = labeled_ord_red_crit_fam.\<G>_set_q q N" for q
       unfolding no_labels.\<G>_set_q_def labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_F_L_q_def by simp
     then have "\<G>_F_L_q q (C,L) \<subseteq> Red_F_q q (labeled_ord_red_crit_fam.\<G>_set_q q N)" if "q \<in> Q" for q
       using that g_in_red unfolding \<G>_F_L_q_def by simp
     then show "(C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N"
       unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q_def
         labeled_ord_red_crit_fam.Red_F_\<G>_q_g_def by blast
   qed
   moreover have ii: \<open>\<exists>C' \<in> (fst ` N). C \<cdot>\<succ> C' \<Longrightarrow>
     (C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N\<close>
   proof -
     assume "\<exists>C' \<in> (fst ` N). C \<cdot>\<succ> C'"
     then obtain C' where c'_in: "C' \<in> (fst ` N)" and c_prec_c': "C \<cdot>\<succ> C'" by blast
     obtain L' where c'_l'_in: "(C',L') \<in> N" using c'_in by auto
     have c'_l'_prec: "(C',L') \<sqsubset> (C,L)"
       using c_prec_c' unfolding Prec_FL_def by (meson UNIV_I compat_equiv_prec)
     have c_in_c'_g: "\<G>_F_q q C \<subseteq> \<G>_F_q q C'" if "q \<in> Q" for q
       using prec_F_grounding[OF that c_prec_c'] by presburger
     then have "\<G>_F_L_q q (C,L) \<subseteq> \<G>_F_L_q q (C',L')" if "q \<in> Q" for q
       unfolding no_labels.\<G>_set_q_def labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_F_L_q_def
       using that by auto
     then have "(C, L) \<in> labeled_ord_red_crit_fam.Red_F_\<G>_q_g q N" if "q \<in> Q" for q
       unfolding labeled_ord_red_crit_fam.Red_F_\<G>_q_g_def using that c'_l'_in c'_l'_prec by blast
     then show "(C, L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N"
       unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q_def by blast
   qed
   moreover have iii: \<open>\<exists>(C', L') \<in> N. (L' \<sqsubset>l L \<and> C \<cdot>\<succeq> C')  \<Longrightarrow>
     (C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N\<close>
   proof -
     assume "\<exists>(C', L') \<in> N. L' \<sqsubset>l L \<and> C \<cdot>\<succeq> C'"
     then obtain C' L' where c'_l'_in: "(C',L') \<in> N" and l'_sub_l: "L' \<sqsubset>l L" and c'_sub_c: "C \<cdot>\<succeq> C'"
       by fast
     have "(C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N" if "C \<cdot>\<succ> C'"
       using that c'_l'_in ii by fastforce
     moreover {
       assume equiv_c_c': "C \<doteq> C'"
       then have equiv_c'_c: "C' \<doteq> C"
         using equiv_F_is_equiv_rel equiv_F_fun_def equiv_class_eq_iff by fastforce
       then have c'_l'_prec: "(C',L') \<sqsubset> (C,L)"
         using l'_sub_l unfolding Prec_FL_def by simp
       have "\<G>_F_q q C = \<G>_F_q q C'" if "q \<in> Q" for q
         using that equiv_F_grounding equiv_c'_c by blast
       then have "\<G>_F_L_q q (C,L) = \<G>_F_L_q q (C',L')" if "q \<in> Q" for q
         unfolding no_labels.\<G>_set_q_def labeled_ord_red_crit_fam.\<G>_set_q_def \<G>_F_L_q_def
         using that by auto
       then have "(C,L) \<in> labeled_ord_red_crit_fam.Red_F_\<G>_q_g q N" if "q \<in> Q" for q
         unfolding labeled_ord_red_crit_fam.Red_F_\<G>_q_g_def using that c'_l'_in c'_l'_prec by blast
       then have "(C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N"
         unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q_def by blast
     }
     ultimately show "(C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N"
       using c'_sub_c unfolding Prec_eq_F_def equiv_F_fun_def equiv_F_is_equiv_rel by blast
   qed
   ultimately show \<open>(C,L) \<in> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N\<close>
     by blast
 qed
 
 end
 
 subsection \<open>Given Clause Architecture\<close>
 
 locale Given_Clause = Prover_Architecture_Basis Bot_F Inf_F Bot_G Q entails_q Inf_G Red_Inf_q
-  Red_F_q \<G>_F_q \<G>_Inf_q l Inf_FL Equiv_F Prec_F Prec_l
+  Red_F_q \<G>_F_q \<G>_Inf_q Inf_FL Equiv_F Prec_F Prec_l
   for
     Bot_F :: "'f set" and
     Inf_F :: "'f inference set" and
     Bot_G :: "'g set" and
     Q :: "'q set" and
     entails_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set \<Rightarrow> bool)" and
     Inf_G :: \<open>'g inference set\<close> and
     Red_Inf_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g inference set)" and
     Red_F_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set)" and
     \<G>_F_q :: "'q \<Rightarrow> 'f \<Rightarrow> 'g set"  and
     \<G>_Inf_q :: "'q \<Rightarrow> 'f inference \<Rightarrow> 'g inference set option" and
-    l :: "'l itself" and
     Inf_FL :: \<open>('f \<times> 'l) inference set\<close> and
     Equiv_F :: "('f \<times> 'f) set" and
     Prec_F :: "'f \<Rightarrow> 'f \<Rightarrow> bool" (infix "\<cdot>\<succ>" 50) and
     Prec_l :: "'l \<Rightarrow> 'l \<Rightarrow> bool" (infix "\<sqsubset>l" 50)
   + fixes
     active :: "'l"
   assumes
     inf_have_premises: "\<iota>F \<in> Inf_F \<Longrightarrow> length (prems_of \<iota>F) > 0" and
     active_minimal: "l2 \<noteq> active \<Longrightarrow> active \<sqsubset>l l2" and
     at_least_two_labels: "\<exists>l2. active \<sqsubset>l l2" and
     inf_never_active: "\<iota> \<in> Inf_FL \<Longrightarrow> snd (concl_of \<iota>) \<noteq> active"
 begin
 
 lemma labeled_inf_have_premises: "\<iota> \<in> Inf_FL \<Longrightarrow> set (prems_of \<iota>) \<noteq> {}"
   using inf_have_premises Inf_FL_to_Inf_F by fastforce
 
 definition active_subset :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "active_subset M = {CL \<in> M. snd CL = active}"
 
 definition non_active_subset :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "non_active_subset M = {CL \<in> M. snd CL \<noteq> active}"
 
 inductive Given_Clause_step :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<Longrightarrow>GC" 50) where
   process: "N1 = N \<union> M \<Longrightarrow> N2 = N \<union> M' \<Longrightarrow> N \<inter> M = {} \<Longrightarrow>
     M \<subseteq>  labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q (N \<union> M') \<Longrightarrow>
     active_subset M' = {} \<Longrightarrow> N1 \<Longrightarrow>GC N2" |
   infer: "N1 = N \<union> {(C,L)} \<Longrightarrow> {(C,L)} \<inter> N = {} \<Longrightarrow> N2 = N \<union> {(C,active)} \<union> M \<Longrightarrow> L \<noteq> active \<Longrightarrow>
     active_subset M = {} \<Longrightarrow>
     no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C} \<subseteq>
       no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N \<union> {(C,active)} \<union> M)) \<Longrightarrow>
     N1 \<Longrightarrow>GC N2"
 
 abbreviation derive :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<rhd>RedL" 50) where
   "derive \<equiv> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive"
 
 lemma one_step_equiv: "N1 \<Longrightarrow>GC N2 \<Longrightarrow> N1 \<rhd>RedL N2"
 proof (cases N1 N2 rule: Given_Clause_step.cases)
   show "N1 \<Longrightarrow>GC N2 \<Longrightarrow> N1 \<Longrightarrow>GC N2" by blast
 next
   fix N M M'
   assume
     gc_step: "N1 \<Longrightarrow>GC N2" and
     n1_is: "N1 = N \<union> M" and
     n2_is: "N2 = N \<union> M'" and
     empty_inter: "N \<inter> M = {}" and
     m_red: "M \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q (N \<union> M')" and
     active_empty: "active_subset M' = {}"
   have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using n1_is n2_is empty_inter m_red by auto
   then show "labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive N1 N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps by blast
 next
   fix N C L M
   assume
     gc_step: "N1 \<Longrightarrow>GC N2" and
     n1_is: "N1 = N \<union> {(C,L)}" and
     not_active: "L \<noteq> active" and
     n2_is: "N2 = N \<union> {(C, active)} \<union> M" and
     empty_inter: "{(C,L)} \<inter> N = {}" and
     active_empty: "active_subset M = {}"
   have "(C, active) \<in> N2" using n2_is by auto
   moreover have "C \<cdot>\<succeq> C" using Prec_eq_F_def equiv_F_is_equiv_rel equiv_class_eq_iff by fastforce
   moreover have "active \<sqsubset>l L" using active_minimal[OF not_active] .
   ultimately have "{(C,L)} \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using red_labeled_clauses by blast
   moreover have "(C,L) \<notin> M \<Longrightarrow> N1 - N2 = {(C,L)}" using n1_is n2_is empty_inter not_active by auto
   moreover have "(C,L) \<in> M \<Longrightarrow> N1 - N2 = {}" using n1_is n2_is by auto
   ultimately have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using empty_red_f_equiv[of N2] by blast
   then show "labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive N1 N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps
     by blast
 qed
 
 abbreviation fair :: "('f \<times> 'l) set llist \<Rightarrow> bool" where
   "fair \<equiv> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.fair"
 
 (* lem:gc-derivations-are-red-derivations *)
 lemma gc_to_red: "chain (\<Longrightarrow>GC) D \<Longrightarrow> chain (\<rhd>RedL) D"
   using one_step_equiv Lazy_List_Chain.chain_mono by blast
 
 lemma (in-) all_ex_finite_set: "(\<forall>(j::nat)\<in>{0..<m}. \<exists>(n::nat). P j n) \<Longrightarrow>
   (\<forall>n1 n2. \<forall>j\<in>{0..<m}. P j n1 \<longrightarrow> P j n2 \<longrightarrow> n1 = n2) \<Longrightarrow> finite {n. \<exists>j \<in> {0..<m}. P j n}" for m P
 proof -
   fix m::nat and P:: "nat \<Rightarrow> nat \<Rightarrow> bool"
   assume
     allj_exn: "\<forall>j\<in>{0..<m}. \<exists>n. P j n" and
     uniq_n: "\<forall>n1 n2. \<forall>j\<in>{0..<m}. P j n1 \<longrightarrow> P j n2 \<longrightarrow> n1 = n2"
   have "{n. \<exists>j \<in> {0..<m}. P j n} = (\<Union>((\<lambda>j. {n. P j n}) ` {0..<m}))" by blast
   then have imp_finite: "(\<forall>j\<in>{0..<m}. finite {n. P j n}) \<Longrightarrow> finite {n. \<exists>j \<in> {0..<m}. P j n}"
     using finite_UN[of "{0..<m}" "\<lambda>j. {n. P j n}"] by simp
   have "\<forall>j\<in>{0..<m}. \<exists>!n. P j n" using allj_exn uniq_n by blast
   then have "\<forall>j\<in>{0..<m}. finite {n. P j n}" by (metis bounded_nat_set_is_finite lessI mem_Collect_eq)
   then show "finite {n. \<exists>j \<in> {0..<m}. P j n}" using imp_finite by simp
 qed
 
 (* lem:fair-gc-derivations *)
 lemma gc_fair: "chain (\<Longrightarrow>GC) D \<Longrightarrow> llength D > 0 \<Longrightarrow> active_subset (lnth D 0) = {} \<Longrightarrow>
   non_active_subset (Liminf_llist D) = {} \<Longrightarrow> fair D"
 proof -
   assume
     deriv: "chain (\<Longrightarrow>GC) D" and
     non_empty: "llength D > 0" and
     init_state: "active_subset (lnth D 0) = {}" and
     final_state: "non_active_subset (Liminf_llist D) = {}"
   show "fair D"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.fair_def
   proof
     fix \<iota>
     assume i_in: "\<iota> \<in> with_labels.Inf_from (Liminf_llist D)"
     have i_in_inf_fl: "\<iota> \<in> Inf_FL" using i_in unfolding with_labels.Inf_from_def by blast
     have "Liminf_llist D = active_subset (Liminf_llist D)"
       using final_state unfolding non_active_subset_def active_subset_def by blast
     then have i_in2: "\<iota> \<in> with_labels.Inf_from (active_subset (Liminf_llist D))" using i_in by simp
     define m where "m = length (prems_of \<iota>)"
     then have m_def_F: "m = length (prems_of (to_F \<iota>))" unfolding to_F_def by simp
     have i_in_F: "to_F \<iota> \<in> Inf_F"
       using i_in Inf_FL_to_Inf_F unfolding with_labels.Inf_from_def to_F_def by blast
     then have m_pos: "m > 0" using m_def_F using inf_have_premises by blast
     have exist_nj: "\<forall>j \<in> {0..<m}. (\<exists>nj. enat (Suc nj) < llength D \<and>
       (prems_of \<iota>)!j \<notin> active_subset (lnth D nj) \<and>
       (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (lnth D k)))"
     proof clarify
       fix j
       assume j_in: "j \<in> {0..<m}"
       then obtain C where c_is: "(C,active) = (prems_of \<iota>)!j"
         using i_in2 unfolding m_def with_labels.Inf_from_def active_subset_def
         by (smt Collect_mem_eq Collect_mono_iff atLeastLessThan_iff nth_mem old.prod.exhaust snd_conv)
       then have "(C,active) \<in> Liminf_llist D"
         using j_in i_in unfolding m_def with_labels.Inf_from_def by force
       then obtain nj where nj_is: "enat nj < llength D" and
         c_in2: "(C,active) \<in> \<Inter> (lnth D ` {k. nj \<le> k \<and> enat k < llength D})"
         unfolding Liminf_llist_def using init_state by blast
       then have c_in3: "\<forall>k. k \<ge> nj \<longrightarrow> enat k < llength D \<longrightarrow> (C,active) \<in> (lnth D k)" by blast
       have nj_pos: "nj > 0" using init_state c_in2 nj_is unfolding active_subset_def by fastforce
       obtain nj_min where nj_min_is: "nj_min = (LEAST nj. enat nj < llength D \<and>
         (C,active) \<in> \<Inter> (lnth D ` {k. nj \<le> k \<and> enat k < llength D}))" by blast
       then have in_allk: "\<forall>k. k \<ge> nj_min \<longrightarrow> enat k < llength D \<longrightarrow> (C,active) \<in> (lnth D k)"
         using c_in3 nj_is c_in2
         by (metis (mono_tags, lifting) INT_E LeastI_ex mem_Collect_eq)
       have njm_smaller_D: "enat nj_min < llength D"
         using nj_min_is
         by (smt LeastI_ex \<open>\<And>thesis. (\<And>nj. \<lbrakk>enat nj < llength D;
           (C, active) \<in> \<Inter> (lnth D ` {k. nj \<le> k \<and> enat k < llength D})\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>)
       have "nj_min > 0"
         using nj_is c_in2 nj_pos nj_min_is
         by (metis (mono_tags, lifting) Collect_empty_eq \<open>(C, active) \<in> Liminf_llist D\<close>
           \<open>Liminf_llist D = active_subset (Liminf_llist D)\<close>
           \<open>\<forall>k\<ge>nj_min. enat k < llength D \<longrightarrow> (C, active) \<in> lnth D k\<close> active_subset_def init_state
           linorder_not_less mem_Collect_eq non_empty zero_enat_def)
       then obtain njm_prec where nj_prec_is: "Suc njm_prec = nj_min" using gr0_conv_Suc by auto
       then have njm_prec_njm: "njm_prec < nj_min" by blast
       then have njm_prec_njm_enat: "enat njm_prec < enat nj_min" by simp
       have njm_prec_smaller_d: "njm_prec < llength D"
         using  HOL.no_atp(15)[OF njm_smaller_D njm_prec_njm_enat] .
       have njm_prec_all_suc: "\<forall>k>njm_prec. enat k < llength D \<longrightarrow> (C, active) \<in> lnth D k"
         using nj_prec_is in_allk by simp
       have notin_njm_prec: "(C, active) \<notin> lnth D njm_prec"
       proof (rule ccontr)
         assume "\<not> (C, active) \<notin> lnth D njm_prec"
         then have absurd_hyp: "(C, active) \<in> lnth D njm_prec" by simp
         have prec_smaller: "enat njm_prec < llength D" using nj_min_is nj_prec_is
           by (smt LeastI_ex Suc_leD \<open>\<And>thesis. (\<And>nj. \<lbrakk>enat nj < llength D;
             (C, active) \<in> \<Inter> (lnth D ` {k. nj \<le> k \<and> enat k < llength D})\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
             enat_ord_simps(1) le_eq_less_or_eq le_less_trans)
         have "(C,active) \<in> \<Inter> (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D})"
           proof -
             {
             fix k
             assume k_in: "njm_prec \<le> k \<and> enat k < llength D"
             have "k = njm_prec \<Longrightarrow> (C,active) \<in> lnth D k" using absurd_hyp by simp
             moreover have "njm_prec < k \<Longrightarrow> (C,active) \<in> lnth D k"
               using nj_prec_is in_allk k_in by simp
             ultimately have "(C,active) \<in> lnth D k" using k_in by fastforce
             }
             then show "(C,active) \<in> \<Inter> (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D})" by blast
           qed
         then have "enat njm_prec < llength D \<and>
           (C,active) \<in> \<Inter> (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D})"
           using prec_smaller by blast
         then show False
           using nj_min_is nj_prec_is Orderings.wellorder_class.not_less_Least njm_prec_njm by blast
       qed
       then have notin_active_subs_njm_prec: "(C, active) \<notin> active_subset (lnth D njm_prec)"
         unfolding active_subset_def by blast
       then show "\<exists>nj. enat (Suc nj) < llength D \<and> (prems_of \<iota>)!j \<notin> active_subset (lnth D nj) \<and>
         (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (lnth D k))"
         using c_is njm_prec_all_suc njm_prec_smaller_d by (metis (mono_tags, lifting)
           active_subset_def mem_Collect_eq nj_prec_is njm_smaller_D snd_conv)
     qed
     define nj_set where "nj_set = {nj. (\<exists>j\<in>{0..<m}. enat (Suc nj) < llength D \<and>
       (prems_of \<iota>)!j \<notin> active_subset (lnth D nj) \<and>
       (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (lnth D k)))}"
     then have nj_not_empty: "nj_set \<noteq> {}"
     proof -
       have zero_in: "0 \<in> {0..<m}" using m_pos by simp
       then obtain n0 where "enat (Suc n0) < llength D" and
         "prems_of \<iota> ! 0 \<notin> active_subset (lnth D n0)" and
         "\<forall>k>n0. enat k < llength D \<longrightarrow> prems_of \<iota> ! 0 \<in> active_subset (lnth D k)"
         using exist_nj by fast
       then have "n0 \<in> nj_set" unfolding nj_set_def using zero_in by blast
       then show "nj_set \<noteq> {}" by auto
     qed
     have nj_finite: "finite nj_set"
       using all_ex_finite_set[OF exist_nj]
       by (metis (no_types, lifting) Suc_ile_eq dual_order.strict_implies_order
         linorder_neqE_nat nj_set_def)
     (* the n below in the n-1 from the pen-and-paper proof *)
     have "\<exists>n \<in> nj_set. \<forall>nj \<in> nj_set. nj \<le> n"
       using nj_not_empty nj_finite using Max_ge Max_in by blast
     then obtain n where n_in: "n \<in> nj_set" and n_bigger: "\<forall>nj \<in> nj_set. nj \<le> n" by blast
     then obtain j0 where j0_in: "j0 \<in> {0..<m}" and suc_n_length: "enat (Suc n) < llength D" and
       j0_notin: "(prems_of \<iota>)!j0 \<notin> active_subset (lnth D n)" and
       j0_allin: "(\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j0 \<in> active_subset (lnth D k))"
       unfolding nj_set_def by blast
     obtain C0 where C0_is: "(prems_of \<iota>)!j0 = (C0,active)" using j0_in
         using i_in2 unfolding m_def with_labels.Inf_from_def active_subset_def
         by (smt Collect_mem_eq Collect_mono_iff atLeastLessThan_iff nth_mem old.prod.exhaust snd_conv)
     then have C0_prems_i: "(C0,active) \<in> set (prems_of \<iota>)" using in_set_conv_nth j0_in m_def by force
     have C0_in: "(C0,active) \<in> (lnth D (Suc n))"
       using C0_is j0_allin suc_n_length by (simp add: active_subset_def)
     have C0_notin: "(C0,active) \<notin> (lnth D n)" using C0_is j0_notin unfolding active_subset_def by simp
     have step_n: "lnth D n \<Longrightarrow>GC lnth D (Suc n)"
       using deriv chain_lnth_rel n_in unfolding nj_set_def by blast
     have "\<exists>N C L M. (lnth D n = N \<union> {(C,L)} \<and> {(C,L)} \<inter> N = {} \<and>
       lnth D (Suc n) = N \<union> {(C,active)} \<union> M \<and> L \<noteq> active \<and>
       active_subset M = {} \<and>
       no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C} \<subseteq>
       no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N \<union> {(C,active)} \<union> M)))"
     proof -
       have proc_or_infer: "(\<exists>N1 N M N2 M'. lnth D n = N1 \<and> lnth D (Suc n) = N2 \<and> N1 = N \<union> M \<and>
          N2 = N \<union> M' \<and> N \<inter> M = {} \<and>
          M \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q (N \<union> M') \<and>
          active_subset M' = {}) \<or>
        (\<exists>N1 N C L N2 M. lnth D n = N1 \<and> lnth D (Suc n) = N2 \<and> N1 = N \<union> {(C, L)} \<and>
          {(C, L)} \<inter> N = {} \<and> N2 = N \<union> {(C, active)} \<union> M \<and>
          L \<noteq> active \<and> active_subset M = {} \<and>
          no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C} \<subseteq>
            no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N \<union> {(C,active)} \<union> M)))"
         using Given_Clause_step.simps[of "lnth D n" "lnth D (Suc n)"] step_n by blast
       show ?thesis
         using C0_in C0_notin proc_or_infer j0_in C0_is
         by (smt Un_iff active_subset_def mem_Collect_eq snd_conv sup_bot.right_neutral)
     qed
     then obtain N M L where inf_from_subs:
       "no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C0} \<subseteq>
       no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N \<union> {(C0,active)} \<union> M))" and
       nth_d_is: "lnth D n = N \<union> {(C0,L)}" and
       suc_nth_d_is: "lnth D (Suc n) = N \<union> {(C0,active)} \<union> M" and
       l_not_active: "L \<noteq> active"
       using C0_in C0_notin j0_in C0_is using active_subset_def by fastforce
     have "j \<in> {0..<m} \<Longrightarrow> (prems_of \<iota>)!j \<noteq> (prems_of \<iota>)!j0 \<Longrightarrow> (prems_of \<iota>)!j \<in> (active_subset N)" for j
     proof -
       fix j
       assume j_in: "j \<in> {0..<m}" and
       j_not_j0: "(prems_of \<iota>)!j \<noteq> (prems_of \<iota>)!j0"
       obtain nj where nj_len: "enat (Suc nj) < llength D" and
         nj_prems: "(prems_of \<iota>)!j \<notin> active_subset (lnth D nj)" and
         nj_greater: "(\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (lnth D k))"
         using exist_nj j_in by blast
       then have "nj \<in> nj_set" unfolding nj_set_def using j_in by blast
       moreover have "nj \<noteq> n"
       proof (rule ccontr)
         assume "\<not> nj \<noteq> n"
         then have "(prems_of \<iota>)!j = (C0,active)"
           using C0_in C0_notin Given_Clause_step.simps[of "lnth D n" "lnth D (Suc n)"] step_n
           by (smt Un_iff Un_insert_right nj_greater nj_prems active_subset_def empty_Collect_eq
             insertE lessI mem_Collect_eq prod.sel(2) suc_n_length)
         then show False using j_not_j0 C0_is by simp
       qed
       ultimately have "nj < n" using n_bigger by force
       then have "(prems_of \<iota>)!j \<in> (active_subset (lnth D n))"
         using nj_greater n_in Suc_ile_eq dual_order.strict_implies_order unfolding nj_set_def by blast
       then show "(prems_of \<iota>)!j \<in> (active_subset N)"
         using nth_d_is l_not_active unfolding active_subset_def by force
     qed
     then have "set (prems_of \<iota>) \<subseteq> active_subset N \<union> {(C0, active)}"
       using C0_prems_i C0_is m_def by (metis Un_iff atLeast0LessThan in_set_conv_nth insertCI lessThan_iff subrelI)
     moreover have "\<not> (set (prems_of \<iota>) \<subseteq> active_subset N - {(C0, active)})"  using C0_prems_i by blast
     ultimately have "\<iota> \<in> with_labels.Inf_from2 (active_subset N) {(C0,active)}"
       using i_in_inf_fl unfolding with_labels.Inf_from2_def with_labels.Inf_from_def by blast
     then have "to_F \<iota> \<in> no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C0}"
       unfolding to_F_def with_labels.Inf_from2_def with_labels.Inf_from_def
         no_labels.Non_ground.Inf_from2_def no_labels.Non_ground.Inf_from_def using Inf_FL_to_Inf_F
       by force
     then have "to_F \<iota> \<in> no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (lnth D (Suc n)))"
       using suc_nth_d_is inf_from_subs by fastforce
     then have "\<forall>q \<in> Q. (\<G>_Inf_q q (to_F \<iota>) \<noteq> None \<and>
       the (\<G>_Inf_q q (to_F \<iota>)) \<subseteq> Red_Inf_q q (\<Union> (\<G>_F_q q ` (fst ` (lnth D (Suc n))))))
       \<or> (\<G>_Inf_q q (to_F \<iota>) = None \<and>
       \<G>_F_q q (concl_of (to_F \<iota>)) \<subseteq> (\<Union> (\<G>_F_q q ` (fst ` (lnth D (Suc n))))) \<union>
         Red_F_q q (\<Union> (\<G>_F_q q ` (fst ` (lnth D (Suc n))))))"
       unfolding to_F_def no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q_def
         no_labels.Red_Inf_\<G>_q_def no_labels.\<G>_set_q_def
       by fastforce
     then have "\<iota> \<in> with_labels.Red_Inf_Q (lnth D (Suc n))"
       unfolding to_F_def with_labels.Red_Inf_Q_def Red_Inf_\<G>_L_q_def \<G>_Inf_L_q_def \<G>_set_L_q_def
         \<G>_F_L_q_def using i_in_inf_fl by auto
     then show "\<iota> \<in>
       labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.inter_red_crit_calculus.Sup_Red_Inf_llist D"
       unfolding
         labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.inter_red_crit_calculus.Sup_Red_Inf_llist_def
       using red_inf_equiv2 suc_n_length by auto
   qed
 qed
 
 (* thm:gc-completeness *)
 theorem gc_complete: "chain (\<Longrightarrow>GC) D \<Longrightarrow> llength D > 0 \<Longrightarrow> active_subset (lnth D 0) = {} \<Longrightarrow>
   non_active_subset (Liminf_llist D) = {} \<Longrightarrow> B \<in> Bot_F \<Longrightarrow>
   no_labels.entails_\<G>_Q (fst ` (lnth D 0)) {B} \<Longrightarrow>
   \<exists>i. enat i < llength D \<and> (\<exists>BL\<in> Bot_FL. BL \<in> (lnth D i))"
 proof -
   fix B
   assume
     deriv: "chain (\<Longrightarrow>GC) D" and
     not_empty_d: "llength D > 0" and
     init_state: "active_subset (lnth D 0) = {}" and
     final_state: "non_active_subset (Liminf_llist D) = {}" and
     b_in: "B \<in> Bot_F" and
     bot_entailed: "no_labels.entails_\<G>_Q (fst ` (lnth D 0)) {B}"
   have labeled_b_in: "(B,active) \<in> Bot_FL" unfolding Bot_FL_def using b_in by simp
   have not_empty_d2: "\<not> lnull D" using not_empty_d by force
   have labeled_bot_entailed: "entails_\<G>_L_Q  (lnth D 0) {(B,active)}"
     using labeled_entailment_lifting bot_entailed by fastforce
   have "fair D" using gc_fair[OF deriv not_empty_d init_state final_state] .
   then have "\<exists>i \<in> {i. enat i < llength D}. \<exists>BL\<in>Bot_FL. BL \<in> lnth D i"
     using labeled_ordered_dynamic_ref_comp labeled_b_in not_empty_d2 gc_to_red[OF deriv]
       labeled_bot_entailed entail_equiv
     unfolding dynamic_refutational_complete_calculus_def
       dynamic_refutational_complete_calculus_axioms_def by blast
   then show ?thesis by blast
 qed
 
 end
 
 subsection \<open>Lazy Given Clause Architecture\<close>
 
 locale Lazy_Given_Clause = Prover_Architecture_Basis Bot_F Inf_F Bot_G Q entails_q Inf_G Red_Inf_q
-  Red_F_q \<G>_F_q \<G>_Inf_q l Inf_FL Equiv_F Prec_F Prec_l
+  Red_F_q \<G>_F_q \<G>_Inf_q Inf_FL Equiv_F Prec_F Prec_l
   for
     Bot_F :: "'f set" and
     Inf_F :: "'f inference set" and
     Bot_G :: "'g set" and
     Q :: "'q set" and
     entails_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set \<Rightarrow> bool)" and
     Inf_G :: \<open>'g inference set\<close> and
     Red_Inf_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g inference set)" and
     Red_F_q :: "'q \<Rightarrow> ('g set \<Rightarrow> 'g set)" and
     \<G>_F_q :: "'q \<Rightarrow> 'f \<Rightarrow> 'g set"  and
     \<G>_Inf_q :: "'q \<Rightarrow> 'f inference \<Rightarrow> 'g inference set option" and
-    l :: "'l itself" and
     Inf_FL :: \<open>('f \<times> 'l) inference set\<close> and
     Equiv_F :: "('f \<times> 'f) set" and
     Prec_F :: "'f \<Rightarrow> 'f \<Rightarrow> bool" (infix "\<cdot>\<succ>" 50) and
     Prec_l :: "'l \<Rightarrow> 'l \<Rightarrow> bool" (infix "\<sqsubset>l" 50)
   + fixes
     active :: "'l"
   assumes
     active_minimal: "l2 \<noteq> active \<Longrightarrow> active \<sqsubset>l l2" and
     at_least_two_labels: "\<exists>l2. active \<sqsubset>l l2" and
     inf_never_active: "\<iota> \<in> Inf_FL \<Longrightarrow> snd (concl_of \<iota>) \<noteq> active"
 begin
 
 definition active_subset :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "active_subset M = {CL \<in> M. snd CL = active}"
 
 definition non_active_subset :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "non_active_subset M = {CL \<in> M. snd CL \<noteq> active}"
 
 inductive Lazy_Given_Clause_step :: "'f inference set \<times> ('f \<times> 'l) set \<Rightarrow>
   'f inference set \<times> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<Longrightarrow>LGC" 50) where
   process: "N1 = N \<union> M \<Longrightarrow> N2 = N \<union> M' \<Longrightarrow> N \<inter> M = {} \<Longrightarrow>
     M \<subseteq>  labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q (N \<union> M') \<Longrightarrow>
     active_subset M' = {} \<Longrightarrow> (T,N1) \<Longrightarrow>LGC (T,N2)" |
   schedule_infer: "T2 = T1 \<union> T' \<Longrightarrow> N1 = N \<union> {(C,L)} \<Longrightarrow> {(C,L)} \<inter> N = {} \<Longrightarrow> N2 = N \<union> {(C,active)} \<Longrightarrow>
     L \<noteq> active \<Longrightarrow> T' = no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C} \<Longrightarrow>
     (T1,N1) \<Longrightarrow>LGC (T2,N2)" |
   compute_infer: "T1 = T2 \<union> {\<iota>} \<Longrightarrow> T2 \<inter> {\<iota>} = {} \<Longrightarrow> N2 = N1 \<union> M \<Longrightarrow> active_subset M = {} \<Longrightarrow>
     \<iota> \<in> no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N1 \<union> M)) \<Longrightarrow>
     (T1,N1) \<Longrightarrow>LGC (T2,N2)" |
   delete_orphans: "T1 = T2 \<union> T' \<Longrightarrow> T2 \<inter> T' = {} \<Longrightarrow>
     T' \<inter> no_labels.Non_ground.Inf_from (fst ` (active_subset N)) = {} \<Longrightarrow> (T1,N) \<Longrightarrow>LGC (T2,N)"
 
 abbreviation derive :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<rhd>RedL" 50) where
   "derive \<equiv> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive"
 
 lemma premise_free_inf_always_from: "\<iota> \<in> Inf_F \<Longrightarrow> length (prems_of \<iota>) = 0 \<Longrightarrow>
   \<iota> \<in> no_labels.Non_ground.Inf_from N"
   unfolding no_labels.Non_ground.Inf_from_def by simp
 
 lemma one_step_equiv: "(T1,N1) \<Longrightarrow>LGC (T2,N2) \<Longrightarrow> N1 \<rhd>RedL N2"
 proof (cases "(T1,N1)" "(T2,N2)" rule: Lazy_Given_Clause_step.cases)
   show "(T1,N1) \<Longrightarrow>LGC (T2,N2) \<Longrightarrow> (T1,N1) \<Longrightarrow>LGC (T2,N2)" by blast
 next
   fix N M M'
   assume
     n1_is: "N1 = N \<union> M" and
     n2_is: "N2 = N \<union> M'" and
     empty_inter: "N \<inter> M = {}" and
     m_red: "M \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q (N \<union> M')"
   have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using n1_is n2_is empty_inter m_red by auto
   then show "N1 \<rhd>RedL N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps by blast
 next
   fix N C L M
   assume
     n1_is: "N1 = N \<union> {(C,L)}" and
     not_active: "L \<noteq> active" and
     n2_is: "N2 = N \<union> {(C, active)}"
   have "(C, active) \<in> N2" using n2_is by auto
   moreover have "C \<cdot>\<succeq> C" using Prec_eq_F_def equiv_F_is_equiv_rel equiv_class_eq_iff by fastforce
   moreover have "active \<sqsubset>l L" using active_minimal[OF not_active] .
   ultimately have "{(C,L)} \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using red_labeled_clauses by blast
   then have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using empty_red_f_equiv[of N2] using n1_is n2_is by blast
   then show "N1 \<rhd>RedL N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps
     by blast
 next
   fix M
   assume
     n2_is: "N2 = N1 \<union> M"
   have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using n2_is by blast
   then show "N1 \<rhd>RedL N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps
     by blast
 next
   assume n2_is: "N2 = N1"
   have "N1 - N2 \<subseteq> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.Red_F_Q N2"
     using n2_is by blast
   then show "N1 \<rhd>RedL N2"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.derive.simps
     by blast
 qed
 
 abbreviation fair :: "('f \<times> 'l) set llist \<Rightarrow> bool" where
   "fair \<equiv> labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.fair"
 
 (* lem:lgc-derivations-are-red-derivations *)
 lemma lgc_to_red: "chain (\<Longrightarrow>LGC) D \<Longrightarrow> chain (\<rhd>RedL) (lmap snd D)"
   using one_step_equiv Lazy_List_Chain.chain_mono by (smt chain_lmap prod.collapse)
 
 (* lem:fair-lgc-derivations *)
 lemma lgc_fair: "chain (\<Longrightarrow>LGC) D \<Longrightarrow> llength D > 0 \<Longrightarrow> active_subset (snd (lnth D 0)) = {} \<Longrightarrow>
   non_active_subset (Liminf_llist (lmap snd D)) = {} \<Longrightarrow> (\<forall>\<iota> \<in> Inf_F. length (prems_of \<iota>) = 0 \<longrightarrow>
   \<iota> \<in> (fst (lnth D 0))) \<Longrightarrow>
   Liminf_llist (lmap fst D) = {} \<Longrightarrow> fair (lmap snd D)"
 proof -
   assume
     deriv: "chain (\<Longrightarrow>LGC) D" and
     non_empty: "llength D > 0" and
     init_state: "active_subset (snd (lnth D 0)) = {}" and
     final_state: "non_active_subset (Liminf_llist (lmap snd D)) = {}" and
     no_prems_init_active: "\<forall>\<iota> \<in> Inf_F. length (prems_of \<iota>) = 0 \<longrightarrow> \<iota> \<in> (fst (lnth D 0))" and
     final_schedule: "Liminf_llist (lmap fst D) = {}"
   show "fair (lmap snd D)"
     unfolding labeled_ord_red_crit_fam.lifted_calc_w_red_crit_family.inter_red_crit_calculus.fair_def
   proof
     fix \<iota>
     assume i_in: "\<iota> \<in> with_labels.Inf_from (Liminf_llist (lmap snd D))"
     have i_in_inf_fl: "\<iota> \<in> Inf_FL" using i_in unfolding with_labels.Inf_from_def by blast
     have "Liminf_llist (lmap snd D) = active_subset (Liminf_llist (lmap snd D))"
       using final_state unfolding non_active_subset_def active_subset_def by blast
     then have i_in2: "\<iota> \<in> with_labels.Inf_from (active_subset (Liminf_llist (lmap snd D)))"
       using i_in by simp
     define m where "m = length (prems_of \<iota>)"
     then have m_def_F: "m = length (prems_of (to_F \<iota>))" unfolding to_F_def by simp
     have i_in_F: "to_F \<iota> \<in> Inf_F"
       using i_in Inf_FL_to_Inf_F unfolding with_labels.Inf_from_def to_F_def by blast
     have exist_nj: "\<forall>j \<in> {0..<m}. (\<exists>nj. enat (Suc nj) < llength D \<and>
       (prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj)) \<and>
       (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k))))"
     proof clarify
       fix j
       assume j_in: "j \<in> {0..<m}"
       then obtain C where c_is: "(C,active) = (prems_of \<iota>)!j"
         using i_in2 unfolding m_def with_labels.Inf_from_def active_subset_def
         by (smt Collect_mem_eq Collect_mono_iff atLeastLessThan_iff nth_mem old.prod.exhaust snd_conv)
       then have "(C,active) \<in> Liminf_llist (lmap snd D)"
         using j_in i_in unfolding m_def with_labels.Inf_from_def by force
       then obtain nj where nj_is: "enat nj < llength D" and
         c_in2: "(C,active) \<in> \<Inter> (snd ` (lnth D ` {k. nj \<le> k \<and> enat k < llength D}))"
         unfolding Liminf_llist_def using init_state by fastforce
       then have c_in3: "\<forall>k. k \<ge> nj \<longrightarrow> enat k < llength D \<longrightarrow> (C,active) \<in> snd (lnth D k)" by blast
       have nj_pos: "nj > 0" using init_state c_in2 nj_is unfolding active_subset_def by fastforce
       obtain nj_min where nj_min_is: "nj_min = (LEAST nj. enat nj < llength D \<and>
         (C,active) \<in> \<Inter> (snd ` (lnth D ` {k. nj \<le> k \<and> enat k < llength D})))" by blast
       then have in_allk: "\<forall>k. k \<ge> nj_min \<longrightarrow> enat k < llength D \<longrightarrow> (C,active) \<in> snd (lnth D k)"
         using c_in3 nj_is c_in2 INT_E LeastI_ex
         by (smt INT_iff INT_simps(10) c_is image_eqI mem_Collect_eq)
       have njm_smaller_D: "enat nj_min < llength D"
         using nj_min_is
         by (smt LeastI_ex \<open>\<And>thesis. (\<And>nj. \<lbrakk>enat nj < llength D;
           (C, active) \<in> \<Inter> (snd ` (lnth D ` {k. nj \<le> k \<and> enat k < llength D}))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>)
       have "nj_min > 0"
         using nj_is c_in2 nj_pos nj_min_is
         by (metis (mono_tags, lifting) active_subset_def emptyE in_allk init_state mem_Collect_eq
           non_empty not_less snd_conv zero_enat_def)
       then obtain njm_prec where nj_prec_is: "Suc njm_prec = nj_min" using gr0_conv_Suc by auto
       then have njm_prec_njm: "njm_prec < nj_min" by blast
       then have njm_prec_njm_enat: "enat njm_prec < enat nj_min" by simp
       have njm_prec_smaller_d: "njm_prec < llength D"
         using  HOL.no_atp(15)[OF njm_smaller_D njm_prec_njm_enat] .
       have njm_prec_all_suc: "\<forall>k>njm_prec. enat k < llength D \<longrightarrow> (C, active) \<in> snd (lnth D k)"
         using nj_prec_is in_allk by simp
       have notin_njm_prec: "(C, active) \<notin> snd (lnth D njm_prec)"
       proof (rule ccontr)
         assume "\<not> (C, active) \<notin> snd (lnth D njm_prec)"
         then have absurd_hyp: "(C, active) \<in> snd (lnth D njm_prec)" by simp
         have prec_smaller: "enat njm_prec < llength D" using nj_min_is nj_prec_is
           by (smt LeastI_ex Suc_leD \<open>\<And>thesis. (\<And>nj. \<lbrakk>enat nj < llength D;
             (C, active) \<in> \<Inter> (snd ` (lnth D ` {k. nj \<le> k \<and> enat k < llength D}))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
             enat_ord_simps(1) le_eq_less_or_eq le_less_trans)
         have "(C,active) \<in> \<Inter> (snd ` (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D}))"
           proof -
             {
             fix k
             assume k_in: "njm_prec \<le> k \<and> enat k < llength D"
             have "k = njm_prec \<Longrightarrow> (C,active) \<in> snd (lnth D k)" using absurd_hyp by simp
             moreover have "njm_prec < k \<Longrightarrow> (C,active) \<in> snd (lnth D k)"
               using nj_prec_is in_allk k_in by simp
             ultimately have "(C,active) \<in> snd (lnth D k)" using k_in by fastforce
             }
             then show "(C,active) \<in> \<Inter> (snd ` (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D}))"
               by blast
           qed
         then have "enat njm_prec < llength D \<and>
           (C,active) \<in> \<Inter> (snd ` (lnth D ` {k. njm_prec \<le> k \<and> enat k < llength D}))"
           using prec_smaller by blast
         then show False
           using nj_min_is nj_prec_is Orderings.wellorder_class.not_less_Least njm_prec_njm by blast
       qed
       then have notin_active_subs_njm_prec: "(C, active) \<notin> active_subset (snd (lnth D njm_prec))"
         unfolding active_subset_def by blast
       then show "\<exists>nj. enat (Suc nj) < llength D \<and> (prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj)) \<and>
         (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))"
         using c_is njm_prec_all_suc njm_prec_smaller_d by (metis (mono_tags, lifting)
           active_subset_def mem_Collect_eq nj_prec_is njm_smaller_D snd_conv)
     qed
     define nj_set where "nj_set = {nj. (\<exists>j\<in>{0..<m}. enat (Suc nj) < llength D \<and>
       (prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj)) \<and>
       (\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k))))}"
     {
       assume m_null: "m = 0"
       then have "enat 0 < llength D \<and> to_F \<iota> \<in> fst (lnth D 0)"
         using no_prems_init_active i_in_F non_empty m_def_F zero_enat_def by auto
       then have "\<exists>n. enat n < llength D \<and> to_F \<iota> \<in> fst (lnth D n)"
         by blast
     }
     moreover {
       assume m_pos: "m > 0"
       have uniq_nj: "j \<in> {0..<m} \<Longrightarrow>
         (enat (Suc nj1) < llength D \<and>
         (prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj1)) \<and>
         (\<forall>k. k > nj1 \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))) \<Longrightarrow>
         (enat (Suc nj2) < llength D \<and>
         (prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj2)) \<and>
         (\<forall>k. k > nj2 \<longrightarrow> enat k < llength D \<longrightarrow> (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))) \<Longrightarrow>
         nj1=nj2"
       proof (clarify, rule ccontr)
         fix j nj1 nj2
         assume "j \<in> {0..<m}" and
           nj1_d: "enat (Suc nj1) < llength D" and
           nj2_d: "enat (Suc nj2) < llength D" and
           nj1_notin: "prems_of \<iota> ! j \<notin> active_subset (snd (lnth D nj1))" and
           k_nj1: "\<forall>k>nj1. enat k < llength D \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (snd (lnth D k))" and
           nj2_notin: "prems_of \<iota> ! j \<notin> active_subset (snd (lnth D nj2))" and
           k_nj2: "\<forall>k>nj2. enat k < llength D \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (snd (lnth D k))" and
           diff_12: "nj1 \<noteq> nj2"
         have "nj1 < nj2 \<Longrightarrow> False"
         proof -
           assume prec_12: "nj1 < nj2"
           have "enat nj2 < llength D" using nj2_d using Suc_ile_eq less_trans by blast
           then have "prems_of \<iota> ! j \<in> active_subset (snd (lnth D nj2))"
             using k_nj1 prec_12 by simp
           then show False using nj2_notin by simp
         qed
         moreover have "nj1 > nj2 \<Longrightarrow> False"
         proof -
           assume prec_21: "nj2 < nj1"
           have "enat nj1 < llength D" using nj1_d using Suc_ile_eq less_trans by blast
           then have "prems_of \<iota> ! j \<in> active_subset (snd (lnth D nj1))"
             using k_nj2 prec_21
             by simp
           then show False using nj1_notin by simp
         qed
         ultimately show False using diff_12 by linarith
       qed
             have nj_not_empty: "nj_set \<noteq> {}"
       proof -
         have zero_in: "0 \<in> {0..<m}" using m_pos by simp
         then obtain n0 where "enat (Suc n0) < llength D" and
           "prems_of \<iota> ! 0 \<notin> active_subset (snd (lnth D n0))" and
           "\<forall>k>n0. enat k < llength D \<longrightarrow> prems_of \<iota> ! 0 \<in> active_subset (snd (lnth D k))"
           using exist_nj by fast
         then have "n0 \<in> nj_set" unfolding nj_set_def using zero_in by blast
         then show "nj_set \<noteq> {}" by auto
       qed
       have nj_finite: "finite nj_set"
         using uniq_nj all_ex_finite_set[OF exist_nj] by (metis (no_types, lifting) Suc_ile_eq
           dual_order.strict_implies_order linorder_neqE_nat nj_set_def)
       have "\<exists>n \<in> nj_set. \<forall>nj \<in> nj_set. nj \<le> n"
         using nj_not_empty nj_finite using Max_ge Max_in by blast
       then obtain n where n_in: "n \<in> nj_set" and n_bigger: "\<forall>nj \<in> nj_set. nj \<le> n" by blast
       then obtain j0 where j0_in: "j0 \<in> {0..<m}" and suc_n_length: "enat (Suc n) < llength D" and
         j0_notin: "(prems_of \<iota>)!j0 \<notin> active_subset (snd (lnth D n))" and
         j0_allin: "(\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
           (prems_of \<iota>)!j0 \<in> active_subset (snd (lnth D k)))"
         unfolding nj_set_def by blast
       obtain C0 where C0_is: "(prems_of \<iota>)!j0 = (C0,active)"
         using j0_in i_in2 unfolding m_def with_labels.Inf_from_def active_subset_def
         by (smt Collect_mem_eq Collect_mono_iff atLeastLessThan_iff nth_mem old.prod.exhaust snd_conv)
       then have C0_prems_i: "(C0,active) \<in> set (prems_of \<iota>)" using in_set_conv_nth j0_in m_def by force
       have C0_in: "(C0,active) \<in> (snd (lnth D (Suc n)))"
         using C0_is j0_allin suc_n_length by (simp add: active_subset_def)
       have C0_notin: "(C0,active) \<notin> (snd (lnth D n))"
         using C0_is j0_notin unfolding active_subset_def by simp
       have step_n: "lnth D n \<Longrightarrow>LGC lnth D (Suc n)"
         using deriv chain_lnth_rel n_in unfolding nj_set_def by blast
       have is_scheduled: "\<exists>T2 T1 T' N1 N C L N2. lnth D n = (T1, N1) \<and> lnth D (Suc n) = (T2, N2) \<and>
         T2 = T1 \<union> T' \<and> N1 = N \<union> {(C, L)} \<and> {(C, L)} \<inter> N = {} \<and> N2 = N \<union> {(C, active)} \<and> L \<noteq> active \<and>
         T' = no_labels.Non_ground.Inf_from2 (fst ` active_subset N) {C}"
         using Lazy_Given_Clause_step.simps[of "lnth D n" "lnth D (Suc n)"] step_n C0_in C0_notin
         unfolding active_subset_def by fastforce
       then obtain T2 T1 T' N1 N L N2 where nth_d_is: "lnth D n = (T1, N1)" and
         suc_nth_d_is: "lnth D (Suc n) = (T2, N2)" and t2_is: "T2 = T1 \<union> T'" and
         n1_is: "N1 = N \<union> {(C0, L)}" "{(C0, L)} \<inter> N = {}" "N2 = N \<union> {(C0, active)}" and
         l_not_active: "L \<noteq> active" and
         tp_is: "T' = no_labels.Non_ground.Inf_from2 (fst ` active_subset N) {C0}"
         using C0_in C0_notin j0_in C0_is using active_subset_def by fastforce
       have "j \<in> {0..<m} \<Longrightarrow> (prems_of \<iota>)!j \<noteq> (prems_of \<iota>)!j0 \<Longrightarrow> (prems_of \<iota>)!j \<in> (active_subset N)"
         for j
       proof -
         fix j
         assume j_in: "j \<in> {0..<m}" and
         j_not_j0: "(prems_of \<iota>)!j \<noteq> (prems_of \<iota>)!j0"
         obtain nj where nj_len: "enat (Suc nj) < llength D" and
           nj_prems: "(prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj))" and
           nj_greater: "(\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow>
             (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))"
           using exist_nj j_in by blast
         then have "nj \<in> nj_set" unfolding nj_set_def using j_in by blast
         moreover have "nj \<noteq> n"
         proof (rule ccontr)
           assume "\<not> nj \<noteq> n"
           then have "(prems_of \<iota>)!j = (C0,active)"
             using C0_in C0_notin Lazy_Given_Clause_step.simps[of "lnth D n" "lnth D (Suc n)"] step_n
               active_subset_def is_scheduled nj_greater nj_prems suc_n_length by auto
           then show False using j_not_j0 C0_is by simp
         qed
         ultimately have "nj < n" using n_bigger by force
         then have "(prems_of \<iota>)!j \<in> (active_subset (snd (lnth D n)))"
           using nj_greater n_in Suc_ile_eq dual_order.strict_implies_order
           unfolding nj_set_def by blast
         then show "(prems_of \<iota>)!j \<in> (active_subset N)"
           using nth_d_is l_not_active n1_is unfolding active_subset_def by force
       qed
       then have prems_i_active: "set (prems_of \<iota>) \<subseteq> active_subset N \<union> {(C0, active)}"
         using C0_prems_i C0_is m_def
         by (metis Un_iff atLeast0LessThan in_set_conv_nth insertCI lessThan_iff subrelI)
       moreover have "\<not> (set (prems_of \<iota>) \<subseteq> active_subset N - {(C0, active)})"  using C0_prems_i by blast
       ultimately have "\<iota> \<in> with_labels.Inf_from2 (active_subset N) {(C0,active)}"
         using i_in_inf_fl prems_i_active unfolding with_labels.Inf_from2_def with_labels.Inf_from_def
         by blast
       then have "to_F \<iota> \<in> no_labels.Non_ground.Inf_from2 (fst ` (active_subset N)) {C0}"
         unfolding to_F_def with_labels.Inf_from2_def with_labels.Inf_from_def
           no_labels.Non_ground.Inf_from2_def no_labels.Non_ground.Inf_from_def
         using Inf_FL_to_Inf_F by force
       then have i_in_t2: "to_F \<iota> \<in> T2" using tp_is t2_is by simp
       have "j \<in> {0..<m} \<Longrightarrow> (\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
         (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))" for j
       proof (cases "j = j0")
         case True
         assume "j = j0"
         then show "(\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
           (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))" using j0_allin by simp
       next
         case False
         assume j_in: "j \<in> {0..<m}" and
           "j \<noteq> j0"
         obtain nj where nj_len: "enat (Suc nj) < llength D" and
           nj_prems: "(prems_of \<iota>)!j \<notin> active_subset (snd (lnth D nj))" and
           nj_greater: "(\<forall>k. k > nj \<longrightarrow> enat k < llength D \<longrightarrow>
             (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))"
           using exist_nj j_in by blast
         then have "nj \<in> nj_set" unfolding nj_set_def using j_in by blast
         then show "(\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
           (prems_of \<iota>)!j \<in> active_subset (snd (lnth D k)))"
           using nj_greater n_bigger by auto
       qed
       then have allj_allk: "(\<forall>c\<in> set (prems_of \<iota>). (\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
         c \<in> active_subset (snd (lnth D k))))"
         using m_def by (metis atLeast0LessThan in_set_conv_nth lessThan_iff)
       have "\<forall>c\<in> set (prems_of \<iota>). snd c = active"
         using prems_i_active unfolding active_subset_def by auto
       then have ex_n_i_in: "\<exists>n. enat (Suc n) < llength D \<and> to_F \<iota> \<in> fst (lnth D (Suc n)) \<and>
         (\<forall>c\<in> set (prems_of \<iota>). snd c = active) \<and>
         (\<forall>c\<in> set (prems_of \<iota>). (\<forall>k. k > n \<longrightarrow> enat k < llength D \<longrightarrow>
           c \<in> active_subset (snd (lnth D k))))"
         using allj_allk i_in_t2 suc_nth_d_is fstI n_in nj_set_def
         by auto
       then have "\<exists>n. enat n < llength D \<and> to_F \<iota> \<in> fst (lnth D n) \<and>
         (\<forall>c\<in> set (prems_of \<iota>). snd c = active) \<and> (\<forall>c\<in> set (prems_of \<iota>). (\<forall>k. k \<ge> n \<longrightarrow>
           enat k < llength D \<longrightarrow> c \<in> active_subset (snd (lnth D k))))"
         by auto
     }
     ultimately obtain n T2 N2 where i_in_suc_n: "to_F \<iota> \<in> fst (lnth D n)" and
       all_prems_active_after: "m > 0 \<Longrightarrow> (\<forall>c\<in> set (prems_of \<iota>). (\<forall>k. k \<ge> n \<longrightarrow> enat k < llength D \<longrightarrow>
                   c \<in> active_subset (snd (lnth D k))))" and
       suc_n_length: "enat n < llength D" and suc_nth_d_is: "lnth D n = (T2, N2)"
       by (metis less_antisym old.prod.exhaust zero_less_Suc)
     then have i_in_t2: "to_F \<iota> \<in> T2" by simp
     have "\<exists>p\<ge>n. enat (Suc p) < llength D \<and> to_F \<iota> \<in> (fst (lnth D p)) \<and> to_F \<iota> \<notin> (fst (lnth D (Suc p)))"
     proof (rule ccontr)
       assume
         contra: "\<not> (\<exists>p\<ge>n. enat (Suc p) < llength D \<and> to_F \<iota> \<in> (fst (lnth D p)) \<and>
                      to_F \<iota> \<notin> (fst (lnth D (Suc p))))"
       then have i_in_suc: "p0 \<ge> n \<Longrightarrow> enat (Suc p0) < llength D \<Longrightarrow> to_F \<iota> \<in> (fst (lnth D p0)) \<Longrightarrow>
         to_F \<iota> \<in> (fst (lnth D (Suc p0)))" for p0
         by blast
       have "p0 \<ge> n \<Longrightarrow> enat p0 < llength D \<Longrightarrow> to_F \<iota> \<in> (fst (lnth D p0))" for p0
       proof (induction rule: nat_induct_at_least)
         case base
         then show ?case using i_in_t2 suc_nth_d_is
         by simp
       next
         case (Suc p0)
         assume p_bigger_n: "n \<le> p0" and
           induct_hyp: "enat p0 < llength D \<Longrightarrow> to_F \<iota> \<in> fst (lnth D p0)" and
           sucsuc_smaller_d: "enat (Suc p0) < llength D"
         have suc_p_bigger_n: "n \<le> p0" using p_bigger_n by simp
         have suc_smaller_d: "enat p0 < llength D"
           using sucsuc_smaller_d Suc_ile_eq dual_order.strict_implies_order by blast
         then have "to_F \<iota> \<in> fst (lnth D p0)" using induct_hyp by blast
         then show ?case using i_in_suc[OF suc_p_bigger_n sucsuc_smaller_d] by blast
       qed
       then have i_in_all_bigger_n: "\<forall>j. j \<ge> n \<and> enat j < llength D \<longrightarrow> to_F \<iota> \<in> (fst (lnth D j))"
         by presburger
       have "llength (lmap fst D) = llength D" by force
       then have "to_F \<iota> \<in> \<Inter> (lnth (lmap fst D) ` {j. n \<le> j \<and> enat j < llength (lmap fst D)})"
         using i_in_all_bigger_n using Suc_le_D by auto
       then have "to_F \<iota> \<in> Liminf_llist (lmap fst D)"
         unfolding Liminf_llist_def using suc_n_length by auto
       then show False using final_schedule by fast
     qed
     then obtain p where p_greater_n: "p \<ge> n" and p_smaller_d: "enat (Suc p) < llength D" and
       i_in_p: "to_F \<iota> \<in> (fst (lnth D p))" and i_notin_suc_p: "to_F \<iota> \<notin> (fst (lnth D (Suc p)))"
       by blast
     have p_neq_n: "Suc p \<noteq> n" using i_notin_suc_p i_in_suc_n by blast
     have step_p: "lnth D p \<Longrightarrow>LGC lnth D (Suc p)" using deriv p_smaller_d chain_lnth_rel by blast
     then have "\<exists>T1 T2 \<iota> N2 N1 M. lnth D p = (T1, N1) \<and> lnth D (Suc p) = (T2, N2) \<and>
       T1 = T2 \<union> {\<iota>} \<and> T2 \<inter> {\<iota>} = {} \<and> N2 = N1 \<union> M \<and> active_subset M = {} \<and>
       \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N1 \<union> M))"
     proof -
       have ci_or_do: "(\<exists>T1 T2 \<iota> N2 N1 M. lnth D p = (T1, N1) \<and> lnth D (Suc p) = (T2, N2) \<and>
         T1 = T2 \<union> {\<iota>} \<and> T2 \<inter> {\<iota>} = {} \<and> N2 = N1 \<union> M \<and> active_subset M = {} \<and>
         \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N1 \<union> M))) \<or>
         (\<exists>T1 T2 T' N. lnth D p = (T1, N) \<and> lnth D (Suc p) = (T2, N) \<and>
         T1 = T2 \<union> T' \<and> T2 \<inter> T' = {} \<and>
         T' \<inter> no_labels.Non_ground.Inf_from (fst ` active_subset N) = {})"
         using Lazy_Given_Clause_step.simps[of "lnth D p" "lnth D (Suc p)"] step_p i_in_p i_notin_suc_p
         by fastforce
       then have p_greater_n_strict: "n < Suc p"
         using suc_nth_d_is p_greater_n i_in_t2 i_notin_suc_p le_eq_less_or_eq by force
       have "m > 0 \<Longrightarrow> j \<in> {0..<m} \<Longrightarrow> (prems_of (to_F \<iota>))!j \<in> (fst ` (active_subset (snd (lnth D p))))"
         for j
       proof -
         fix j
         assume
           m_pos: "m > 0" and
           j_in: "j \<in> {0..<m}"
         then have "(prems_of \<iota>)!j \<in> (active_subset (snd (lnth D p)))"
           using all_prems_active_after[OF m_pos] p_smaller_d m_def p_greater_n p_neq_n
           by (meson Suc_ile_eq atLeastLessThan_iff dual_order.strict_implies_order nth_mem
             p_greater_n_strict)
         then have "fst ((prems_of \<iota>)!j) \<in> (fst ` (active_subset (snd (lnth D p))))"
           by blast
         then show "(prems_of (to_F \<iota>))!j \<in> (fst ` (active_subset (snd (lnth D p))))"
         unfolding to_F_def using j_in m_def by simp
       qed
       then have prems_i_active_p: "m > 0 \<Longrightarrow>
         to_F \<iota> \<in> no_labels.Non_ground.Inf_from (fst ` active_subset (snd (lnth D p)))"
         using i_in_F unfolding no_labels.Non_ground.Inf_from_def
         by (smt atLeast0LessThan in_set_conv_nth lessThan_iff m_def_F mem_Collect_eq subsetI)
       have "m = 0 \<Longrightarrow> (\<exists>T1 T2 \<iota> N2 N1 M. lnth D p = (T1, N1) \<and> lnth D (Suc p) = (T2, N2) \<and>
         T1 = T2 \<union> {\<iota>} \<and> T2 \<inter> {\<iota>} = {} \<and> N2 = N1 \<union> M \<and> active_subset M = {} \<and>
         \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N1 \<union> M)))"
         using ci_or_do premise_free_inf_always_from[of "to_F \<iota>" "fst ` active_subset _", OF i_in_F]
           m_def i_in_p i_notin_suc_p m_def_F by auto
       then show "(\<exists>T1 T2 \<iota> N2 N1 M. lnth D p = (T1, N1) \<and> lnth D (Suc p) = (T2, N2) \<and>
         T1 = T2 \<union> {\<iota>} \<and> T2 \<inter> {\<iota>} = {} \<and> N2 = N1 \<union> M \<and> active_subset M = {} \<and>
         \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (N1 \<union> M)))"
         using ci_or_do i_in_p i_notin_suc_p prems_i_active_p unfolding active_subset_def
         by force
     qed
     then obtain T1p T2p N1p N2p Mp where  "lnth D p = (T1p, N1p)" and
       suc_p_is: "lnth D (Suc p) = (T2p, N2p)" and "T1p = T2p \<union> {to_F \<iota>}" and "T2p \<inter> {to_F \<iota>} = {}" and
       n2p_is: "N2p = N1p \<union> Mp"and "active_subset Mp = {}" and
       i_in_red_inf: "to_F \<iota> \<in> no_labels.empty_ord_lifted_calc_w_red_crit_family.Red_Inf_Q
         (fst ` (N1p \<union> Mp))"
       using i_in_p i_notin_suc_p by fastforce
     have "to_F \<iota> \<in> no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q (fst ` (snd (lnth D (Suc p))))"
       using i_in_red_inf suc_p_is n2p_is by fastforce
     then have "\<forall>q \<in> Q. (\<G>_Inf_q q (to_F \<iota>) \<noteq> None \<and>
       the (\<G>_Inf_q q (to_F \<iota>)) \<subseteq> Red_Inf_q q (\<Union> (\<G>_F_q q ` (fst ` (snd (lnth D (Suc p)))))))
       \<or> (\<G>_Inf_q q (to_F \<iota>) = None \<and>
       \<G>_F_q q (concl_of (to_F \<iota>)) \<subseteq> (\<Union> (\<G>_F_q q ` (fst ` (snd (lnth D (Suc p)))))) \<union>
         Red_F_q q (\<Union> (\<G>_F_q q ` (fst ` (snd (lnth D (Suc p)))))))"
       unfolding to_F_def no_labels.lifted_calc_w_red_crit_family.Red_Inf_Q_def
         no_labels.Red_Inf_\<G>_q_def no_labels.\<G>_set_q_def
       by fastforce
     then have "\<iota> \<in> with_labels.Red_Inf_Q (snd (lnth D (Suc p)))"
       unfolding to_F_def with_labels.Red_Inf_Q_def Red_Inf_\<G>_L_q_def \<G>_Inf_L_q_def \<G>_set_L_q_def
         \<G>_F_L_q_def using i_in_inf_fl by auto
     then show "\<iota> \<in> labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.inter_red_crit_calculus.Sup_Red_Inf_llist (lmap snd D)"
       unfolding
         labeled_ord_red_crit_fam.empty_ord_lifted_calc_w_red_crit_family.inter_red_crit_calculus.Sup_Red_Inf_llist_def
       using red_inf_equiv2 suc_n_length p_smaller_d by auto
   qed
 qed
 
 (* thm:lgc-completeness *)
 theorem lgc_complete: "chain (\<Longrightarrow>LGC) D \<Longrightarrow> llength D > 0 \<Longrightarrow> active_subset (snd (lnth D 0)) = {} \<Longrightarrow>
   non_active_subset (Liminf_llist (lmap snd D)) = {} \<Longrightarrow>
   (\<forall>\<iota> \<in> Inf_F. length (prems_of \<iota>) = 0 \<longrightarrow> \<iota> \<in> (fst (lnth D 0))) \<Longrightarrow>
   Liminf_llist (lmap fst D) = {} \<Longrightarrow> B \<in> Bot_F \<Longrightarrow> no_labels.entails_\<G>_Q (fst ` (snd (lnth D 0))) {B} \<Longrightarrow>
   \<exists>i. enat i < llength D \<and> (\<exists>BL\<in> Bot_FL. BL \<in> (snd (lnth D i)))"
 proof -
   fix B
   assume
     deriv: "chain (\<Longrightarrow>LGC) D" and
     not_empty_d: "llength D > 0" and
     init_state: "active_subset (snd (lnth D 0)) = {}" and
     final_state: "non_active_subset (Liminf_llist (lmap snd D)) = {}" and
     no_prems_init_active: "\<forall>\<iota> \<in> Inf_F. length (prems_of \<iota>) = 0 \<longrightarrow> \<iota> \<in> fst (lnth D 0)" and
     final_schedule: "Liminf_llist (lmap fst D) = {}" and
     b_in: "B \<in> Bot_F" and
     bot_entailed: "no_labels.entails_\<G>_Q (fst ` (snd (lnth D 0))) {B}"
   have labeled_b_in: "(B,active) \<in> Bot_FL" unfolding Bot_FL_def using b_in by simp
   have not_empty_d2: "\<not> lnull (lmap snd D)" using not_empty_d by force
   have simp_snd_lmap: "lnth (lmap snd D) 0 = snd (lnth D 0)"
     using lnth_lmap[of 0 D snd] not_empty_d by (simp add: zero_enat_def)
   have labeled_bot_entailed: "entails_\<G>_L_Q  (snd (lnth D 0)) {(B,active)}"
     using labeled_entailment_lifting bot_entailed by fastforce
   have "fair (lmap snd D)"
     using lgc_fair[OF deriv not_empty_d init_state final_state no_prems_init_active final_schedule] .
   then have "\<exists>i \<in> {i. enat i < llength D}. \<exists>BL\<in>Bot_FL. BL \<in> (snd (lnth D i))"
     using labeled_ordered_dynamic_ref_comp labeled_b_in not_empty_d2 lgc_to_red[OF deriv]
       labeled_bot_entailed entail_equiv simp_snd_lmap
     unfolding dynamic_refutational_complete_calculus_def
       dynamic_refutational_complete_calculus_axioms_def
     by (metis (mono_tags, lifting) llength_lmap lnth_lmap mem_Collect_eq)
   then show ?thesis by blast
 qed
 
 end
 
 end