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
@@ -1,356 +1,362 @@
 (*  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 = no_labels: 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
     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>
 
 abbreviation Bot_FL :: \<open>('f \<times> 'l) set\<close> where
   \<open>Bot_FL \<equiv> Bot_F \<times> UNIV\<close>
 
 abbreviation \<G>_F_L :: \<open>('f \<times> 'l) \<Rightarrow> 'g set\<close> where
   \<open>\<G>_F_L CL \<equiv> \<G>_F (fst CL)\<close>
 
 abbreviation \<G>_Inf_L :: \<open>('f \<times> 'l) inference \<Rightarrow> 'g inference set option\<close> where
   \<open>\<G>_Inf_L \<iota>\<^sub>F\<^sub>L \<equiv> \<G>_Inf (to_F \<iota>\<^sub>F\<^sub>L)\<close>
 
 (* lem:labeled-grounding-function *)
 sublocale standard_lifting Bot_FL Inf_FL Bot_G Inf_G "(\<Turnstile>G)" Red_Inf_G Red_F_G \<G>_F_L \<G>_Inf_L
 proof
   show "Bot_FL \<noteq> {}"
     using no_labels.Bot_F_not_empty by simp
 next
   show "B \<in> Bot_FL \<Longrightarrow> \<G>_F_L B \<noteq> {}" for B
     using no_labels.Bot_map_not_empty by auto
 next
   show "B \<in> Bot_FL \<Longrightarrow> \<G>_F_L B \<subseteq> Bot_G" for B
     using no_labels.Bot_map by force
 next
   fix CL
   show "\<G>_F_L CL \<inter> Bot_G \<noteq> {} \<longrightarrow> CL \<in> Bot_FL"
     using no_labels.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 to_F_def using no_labels.inf_map Inf_FL_to_Inf_F by fastforce
 qed
 
 sublocale 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 Cl Cl'. False"
   by unfold_locales simp+
 
 notation 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"
   by simp
 
 lemma red_inf_impl: "\<iota> \<in> Red_Inf_\<G> NL \<Longrightarrow> to_F \<iota> \<in> no_labels.Red_Inf_\<G> (fst ` NL)"
   unfolding Red_Inf_\<G>_def no_labels.Red_Inf_\<G>_def using Inf_FL_to_Inf_F by (auto simp: to_F_def)
 
 (* lem:labeled-saturation *)
 lemma labeled_saturation_lifting: "saturated NL \<Longrightarrow> no_labels.saturated (fst ` NL)"
   unfolding saturated_def no_labels.saturated_def Inf_from_def no_labels.Inf_from_def
 proof clarify
   fix \<iota>
   assume
     subs_Red_Inf: "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> 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 = (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 i_prems unfolding Lli_def by (metis nth_mem someI_ex DomainE Domain_fst subset_eq)
   define Ll where "Ll = 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> 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> no_labels.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:
   assumes static:
     "static_refutational_complete_calculus Bot_F Inf_F (\<Turnstile>\<G>) no_labels.Red_Inf_\<G> no_labels.Red_F_\<G>"
   shows "static_refutational_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<G>L) Red_Inf_\<G> Red_F_\<G>"
 proof
   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>saturated Nl\<close> and
     Nl_entails_Bl: \<open>Nl \<Turnstile>\<G>L {Bl}\<close>
   define B where "B = fst Bl"
   have B_in: "B \<in> Bot_F" using Bl_in B_def SigmaE by force
   define N where "N = fst ` Nl"
   have N_sat: "no_labels.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
     using static[unfolded static_refutational_complete_calculus_def
         static_refutational_complete_calculus_axioms_def] 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" 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" 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
 
 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 Q Inf_G_q entails_q Red_Inf_q Red_F_q Bot_F \<G>_F_q \<G>_Inf_q "\<lambda>g Cl Cl'. False"
   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_q :: "'q \<Rightarrow> '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
     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>
 
 abbreviation Bot_FL :: \<open>('f \<times> 'l) set\<close> where
   \<open>Bot_FL \<equiv> Bot_F \<times> UNIV\<close>
 
 abbreviation \<G>_F_L_q :: \<open>'q \<Rightarrow> ('f \<times> 'l) \<Rightarrow> 'g set\<close> where
   \<open>\<G>_F_L_q q CL \<equiv> \<G>_F_q q (fst CL)\<close>
 
 abbreviation \<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 \<equiv> \<G>_Inf_q q (to_F \<iota>\<^sub>F\<^sub>L)\<close>
 
 abbreviation \<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))}"
 
 abbreviation Red_Inf_\<G>_L_Q :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) inference set" where
   "Red_Inf_\<G>_L_Q N \<equiv> (\<Inter>q \<in> Q. Red_Inf_\<G>_L_q q N)"
 
 abbreviation 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)"
 
 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_q q) (Red_Inf_q q)
     (Red_F_q q) (\<G>_F_q q) (\<G>_Inf_q q) (\<lambda>g Cl Cl'. False) 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_q q) (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_q q"
     "Red_Inf_q q" "Red_F_q q" "\<G>_F_q q" "\<G>_Inf_q q" "\<lambda>g Cl Cl'. False" Inf_FL
     using lifting_q[OF q_in] .
   have "\<G>_Inf_L_q q = q_lifting.\<G>_Inf_L"
     unfolding to_F_def q_lifting.to_F_def by simp
   then show ?thesis
     using q_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_q q) (Red_Inf_q q)
     (Red_F_q q) (\<G>_F_L_q q) (\<G>_Inf_L_q q) (\<lambda>g Cl Cl'. False)"
 proof -
   interpret standard_q_lifting: standard_lifting Bot_FL Inf_FL Bot_G "Inf_G_q q" "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 (\<lambda>Cl Cl'. False) UNIV"
     by (simp add: minimal_element.intro po_on_def transp_onI wfp_on_imp_irreflp_on)
   then show ?thesis
     using standard_q_lifting.standard_lifting_axioms
     by (simp add: lifting_with_wf_ordering_family_axioms_def lifting_with_wf_ordering_family_def)
 qed
 
 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. False \<and> D \<in> \<G>_F_L_q q E)}"
 
 abbreviation Red_F_\<G>_empty_L :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set" where
   "Red_F_\<G>_empty_L N \<equiv> (\<Inter>q \<in> Q. Red_F_\<G>_empty_L_q q N)"
 
 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_q q" "Red_Inf_q q" "Red_F_q q" "\<G>_F_L_q q" "\<G>_Inf_L_q q" "\<lambda>g Cl Cl'. False"
     using ord_fam_lifted_q[OF q_in] .
   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 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 by simp
   ultimately show ?thesis
     using ord_q_lifting.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)"
   using all_lifted_red_crit calculus_with_red_crit_def q_in by blast
 
 sublocale consequence_relation_family Bot_FL Q entails_\<G>_L_q
   using all_lifted_cons_rel by (simp add: consequence_relation_family.intro no_labels.Q_nonempty)
 
 sublocale 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 consequence_relation_family_axioms]
   by (simp add: all_lifted_red_crit calculus_with_red_crit_family_axioms_def no_labels.Q_nonempty)
 
+lemma in_Inf_FL_imp_to_F_in_Inf_F: "\<iota> \<in> Inf_FL \<Longrightarrow> to_F \<iota> \<in> Inf_F"
+  by (simp add: Inf_FL_to_Inf_F to_F_def)
+
+lemma in_Inf_from_imp_to_F_in_Inf_from: "\<iota> \<in> Inf_from N \<Longrightarrow> to_F \<iota> \<in> no_labels.Inf_from (fst ` N)"
+  unfolding Inf_from_def no_labels.Inf_from_def to_F_def by (auto intro: Inf_FL_to_Inf_F)
+
 notation no_labels.entails_\<G>_Q (infix "\<Turnstile>\<inter>\<G>" 50)
 
 abbreviation entails_\<G>_L_Q :: "('f \<times> 'l) set \<Rightarrow> ('f \<times> 'l) set \<Rightarrow> bool" (infix "\<Turnstile>\<inter>\<G>L" 50) where
   "(\<Turnstile>\<inter>\<G>L) \<equiv> entails_Q"
 
 lemmas entails_\<G>_L_Q_def = entails_Q_def
 
 (* lem:labeled-consequence-intersection *)
 lemma labeled_entailment_lifting: "NL1 \<Turnstile>\<inter>\<G>L NL2 \<longleftrightarrow> fst ` NL1 \<Turnstile>\<inter>\<G> fst ` NL2"
   unfolding no_labels.entails_\<G>_Q_def entails_\<G>_L_Q_def by force
 
 lemma red_inf_impl: "\<iota> \<in> Red_Inf_Q NL \<Longrightarrow> to_F \<iota> \<in> no_labels.Red_Inf_\<G>_Q (fst ` NL)"
   unfolding no_labels.Red_Inf_\<G>_Q_def Red_Inf_Q_def
 proof clarify
   fix X Xa q
   assume
     q_in: "q \<in> Q" and
     i_in_inter: "\<iota> \<in> (\<Inter>q \<in> Q. Red_Inf_\<G>_L_q q NL)"
   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)))"
     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: "saturated NL \<Longrightarrow> no_labels.saturated (fst ` NL)"
   unfolding saturated_def no_labels.saturated_def Inf_from_def no_labels.Inf_from_def
 proof clarify
   fix \<iota>F
   assume
     labeled_sat: "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> 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 = (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 iF_prems nth_mem someI_ex unfolding Lli_def by (metis DomainE Domain_fst subset_eq)
   define Ll where "Ll = 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> 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.Red_Inf_\<G>_Q (fst ` NL)"
     by (auto intro: red_inf_impl)
 qed
 
 (* thm:labeled-static-ref-compl-intersection *)
 theorem labeled_static_ref:
   assumes calc: "static_refutational_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) no_labels.Red_Inf_\<G>_Q
     no_labels.Red_F_\<G>_empty"
   shows "static_refutational_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<inter>\<G>L) Red_Inf_Q Red_F_Q"
 proof
   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>saturated Nl\<close> and
     Nl_entails_Bl: \<open>Nl \<Turnstile>\<inter>\<G>L {Bl}\<close>
   define B where "B = fst Bl"
   have B_in: "B \<in> Bot_F" using Bl_in B_def SigmaE by force
   define N where "N = fst ` Nl"
   have N_sat: "no_labels.saturated N"
     using N_def Nl_sat labeled_family_saturation_lifting by blast
   have N_entails_B: "N \<Turnstile>\<inter>\<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
       calc[unfolded static_refutational_complete_calculus_def
         static_refutational_complete_calculus_axioms_def]
     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" 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" 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
 
 end
 
 end