# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1556141343 -3600
# Node ID da497279f492b570f9760ba6396dc0d785913759
# Parent  49a65e3f04c94ea12d254b95d18fa3355769f644
getting rid of most apply steps

diff -r 49a65e3f04c9 -r da497279f492 src/HOL/ex/Tarski.thy
--- a/src/HOL/ex/Tarski.thy	Mon Apr 22 09:33:55 2019 +0000
+++ b/src/HOL/ex/Tarski.thy	Wed Apr 24 22:29:03 2019 +0100
@@ -100,24 +100,18 @@
   assumes cl_co: "cl \<in> CompleteLattice"
 
 sublocale CL < po?: PO
-    apply (simp_all add: A_def r_def)
-  apply unfold_locales
-  using cl_co unfolding CompleteLattice_def
-  apply auto
-  done
+  unfolding A_def r_def
+  using CompleteLattice_def PO.intro cl_co by fastforce
 
 locale CLF = S +
   fixes f :: "'a \<Rightarrow> 'a"
     and P :: "'a set"
-  assumes f_cl:  "(cl, f) \<in> CLF_set" (*was the equivalent "f \<in> CLF_set``{cl}"*)
+  assumes f_cl:  "(cl, f) \<in> CLF_set" 
   defines P_def: "P \<equiv> fix f A"
 
 sublocale CLF < cl?: CL
-    apply (simp_all add: A_def r_def)
-  apply unfold_locales
-  using f_cl unfolding CLF_set_def
-  apply auto
-  done
+  unfolding A_def r_def CL_def
+  using CLF_set_def f_cl by blast
 
 locale Tarski = CLF +
   fixes Y :: "'a set"
@@ -136,11 +130,10 @@
 begin
 
 lemma dual: "PO (dual cl)"
-  apply unfold_locales
-  using cl_po
-  unfolding PartialOrder_def dual_def
-  apply auto
-  done
+proof
+  show "dual cl \<in> PartialOrder"
+  using cl_po unfolding PartialOrder_def dual_def by auto
+qed
 
 lemma PO_imp_refl_on [simp]: "refl_on A r"
   using cl_po by (simp add: PartialOrder_def A_def r_def)
@@ -163,19 +156,20 @@
 lemma monotoneE: "\<lbrakk>monotone f A r; x \<in> A; y \<in> A; (x, y) \<in> r\<rbrakk> \<Longrightarrow> (f x, f y) \<in> r"
   by (simp add: monotone_def)
 
-lemma po_subset_po: "S \<subseteq> A \<Longrightarrow> \<lparr>pset = S, order = induced S r\<rparr> \<in> PartialOrder"
-  apply (simp add: PartialOrder_def)
-  apply auto
-    \<comment> \<open>refl\<close>
-    apply (simp add: refl_on_def induced_def)
-    apply (blast intro: reflE)
-    \<comment> \<open>antisym\<close>
-   apply (simp add: antisym_def induced_def)
-   apply (blast intro: antisymE)
-    \<comment> \<open>trans\<close>
-  apply (simp add: trans_def induced_def)
-  apply (blast intro: transE)
-  done
+lemma po_subset_po: 
+  assumes "S \<subseteq> A" shows "\<lparr>pset = S, order = induced S r\<rparr> \<in> PartialOrder"
+proof -
+  have "refl_on S (induced S r)"
+    using \<open>S \<subseteq> A\<close> by (auto simp: refl_on_def induced_def intro: reflE)
+  moreover
+  have "antisym (induced S r)"
+    by (auto simp add: antisym_def induced_def intro: antisymE)
+  moreover
+  have "trans (induced S r)"
+    by (auto simp add: trans_def induced_def intro: transE)
+  ultimately show ?thesis
+    by (simp add: PartialOrder_def)
+qed
 
 lemma indE: "\<lbrakk>(x, y) \<in> induced S r; S \<subseteq> A\<rbrakk> \<Longrightarrow> (x, y) \<in> r"
   by (simp add: induced_def)
@@ -206,17 +200,15 @@
   using cl_po by (simp add: PartialOrder_def dual_def)
 
 lemma Rdual:
-  "\<forall>S. (S \<subseteq> A \<longrightarrow> (\<exists>L. isLub S \<lparr>pset = A, order = r\<rparr> L))
-    \<Longrightarrow> \<forall>S. S \<subseteq> A \<longrightarrow> (\<exists>G. isGlb S \<lparr>pset = A, order = r\<rparr> G)"
-  apply safe
-  apply (rule_tac x = "lub {y. y \<in> A \<and> (\<forall>k \<in> S. (y, k) \<in> r)} \<lparr>pset = A, order = r\<rparr>" in exI)
-  apply (drule_tac x = "{y. y \<in> A \<and> (\<forall>k \<in> S. (y, k) \<in> r)}" in spec)
-  apply (drule mp)
-   apply fast
-  apply (simp add: isLub_lub isGlb_def)
-  apply (simp add: isLub_def)
-  apply blast
-  done
+  assumes major: "\<And>S. S \<subseteq> A \<Longrightarrow> \<exists>L. isLub S po L" and "S \<subseteq> A" and "A = pset po"
+  shows "\<exists>G. isGlb S po G"
+proof
+  show "isGlb S po (lub {y \<in> A. \<forall>k\<in>S. (y, k) \<in> order po} po)"
+    using major [of "{y. y \<in> A \<and> (\<forall>k \<in> S. (y, k) \<in> order po)}"] \<open>S \<subseteq> A\<close> \<open>A = pset po\<close>
+    apply (simp add: isLub_lub isGlb_def)
+    apply (auto simp add: isLub_def)
+    done
+qed
 
 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
   by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)
@@ -229,10 +221,6 @@
 
 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
 
-(*declare CL_imp_PO [THEN PO.PO_imp_refl, simp]
-declare CL_imp_PO [THEN PO.PO_imp_sym, simp]
-declare CL_imp_PO [THEN PO.PO_imp_trans, simp]*)
-
 context CL
 begin
 
@@ -256,53 +244,36 @@
 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
   using cl_co
   apply (simp add: CompleteLattice_def dual_def)
-  apply (fold dual_def)
-  apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric] dualPO)
+  apply (simp add: dualPO flip: dual_def isLub_dual_isGlb isGlb_dual_isLub)
   done
 
 context PO
 begin
 
-lemma dualA_iff: "pset (dual cl) = pset cl"
+lemma dualA_iff [simp]: "pset (dual cl) = pset cl"
   by (simp add: dual_def)
 
-lemma dualr_iff: "(x, y) \<in> (order (dual cl)) \<longleftrightarrow> (y, x) \<in> order cl"
+lemma dualr_iff [simp]: "(x, y) \<in> (order (dual cl)) \<longleftrightarrow> (y, x) \<in> order cl"
   by (simp add: dual_def)
 
 lemma monotone_dual:
   "monotone f (pset cl) (order cl) \<Longrightarrow> monotone f (pset (dual cl)) (order(dual cl))"
-  by (simp add: monotone_def dualA_iff dualr_iff)
+  by (simp add: monotone_def)
 
 lemma interval_dual: "\<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> interval r x y = interval (order(dual cl)) y x"
-  apply (simp add: interval_def dualr_iff)
-  apply (fold r_def)
-  apply fast
-  done
-
-lemma trans: "(x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r"
-  using cl_po
-  apply (auto simp add: PartialOrder_def r_def)
-  unfolding trans_def
-  apply blast
-  done
+  unfolding interval_def dualr_iff by (auto simp flip: r_def)
 
 lemma interval_not_empty: "interval r a b \<noteq> {} \<Longrightarrow> (a, b) \<in> r"
-  by (simp add: interval_def) (use trans in blast)
+  by (simp add: interval_def) (use transE in blast)
 
 lemma interval_imp_mem: "x \<in> interval r a b \<Longrightarrow> (a, x) \<in> r"
   by (simp add: interval_def)
 
 lemma left_in_interval: "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}\<rbrakk> \<Longrightarrow> a \<in> interval r a b"
-  apply (simp (no_asm_simp) add: interval_def)
-  apply (simp add: interval_not_empty)
-  apply (simp add: reflE)
-  done
+  using interval_def interval_not_empty reflE by fastforce
 
 lemma right_in_interval: "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}\<rbrakk> \<Longrightarrow> b \<in> interval r a b"
-  apply (simp (no_asm_simp) add: interval_def)
-  apply (simp add: interval_not_empty)
-  apply (simp add: reflE)
-  done
+  by (simp add: A_def PO.dual PO.left_in_interval PO_axioms interval_dual)
 
 end
 
@@ -318,12 +289,11 @@
   by (simp add: sublattice_def A_def r_def)
 
 lemma (in CL) dual: "CL (dual cl)"
-  apply unfold_locales
+proof
+  show "dual cl \<in> CompleteLattice"
   using cl_co
-  unfolding CompleteLattice_def
-  apply (simp add: dualPO isGlb_dual_isLub [symmetric] isLub_dual_isGlb [symmetric] dualA_iff)
-  done
-
+  by (simp add: CompleteLattice_def dualPO flip: isGlb_dual_isLub isLub_dual_isGlb)
+qed
 
 subsection \<open>lub\<close>
 
@@ -333,47 +303,46 @@
 lemma lub_unique: "\<lbrakk>S \<subseteq> A; isLub S cl x; isLub S cl L\<rbrakk> \<Longrightarrow> x = L"
   by (rule antisymE) (auto simp add: isLub_def r_def)
 
-lemma lub_upper: "\<lbrakk>S \<subseteq> A; x \<in> S\<rbrakk> \<Longrightarrow> (x, lub S cl) \<in> r"
-  apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-  apply (unfold lub_def least_def)
-  apply (rule some_equality [THEN ssubst])
-    apply (simp add: isLub_def)
-   apply (simp add: lub_unique A_def isLub_def)
-  apply (simp add: isLub_def r_def)
-  done
+lemma lub_upper: 
+  assumes "S \<subseteq> A" "x \<in> S" shows "(x, lub S cl) \<in> r"
+proof -
+  obtain L where "isLub S cl L"
+    using CL_imp_ex_isLub \<open>S \<subseteq> A\<close> by auto
+  then show ?thesis
+    by (metis assms(2) isLub_def isLub_lub r_def)
+qed
 
-lemma lub_least: "\<lbrakk>S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x, L) \<in> r\<rbrakk> \<Longrightarrow> (lub S cl, L) \<in> r"
-  apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-  apply (unfold lub_def least_def)
-  apply (rule_tac s=x in some_equality [THEN ssubst])
-    apply (simp add: isLub_def)
-   apply (simp add: lub_unique A_def isLub_def)
-  apply (simp add: isLub_def r_def A_def)
-  done
+lemma lub_least:
+  assumes "S \<subseteq> A" and L: "L \<in> A" "\<forall>x \<in> S. (x, L) \<in> r" shows "(lub S cl, L) \<in> r"
+proof -
+  obtain L' where "isLub S cl L'"
+    using CL_imp_ex_isLub \<open>S \<subseteq> A\<close> by auto
+  then show ?thesis
+    by (metis A_def L isLub_def isLub_lub r_def)
+qed
 
-lemma lub_in_lattice: "S \<subseteq> A \<Longrightarrow> lub S cl \<in> A"
-  apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-  apply (unfold lub_def least_def)
-  apply (subst some_equality)
-    apply (simp add: isLub_def)
-   prefer 2 apply (simp add: isLub_def A_def)
-  apply (simp add: lub_unique A_def isLub_def)
-  done
+lemma lub_in_lattice:
+  assumes "S \<subseteq> A" shows "lub S cl \<in> A"
+proof -
+  obtain L where "isLub S cl L"
+    using CL_imp_ex_isLub \<open>S \<subseteq> A\<close> by auto
+  then show ?thesis
+    by (metis A_def isLub_def isLub_lub)
+qed
 
 lemma lubI:
-  "\<lbrakk>S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x, L) \<in> r;
-    \<forall>z \<in> A. (\<forall>y \<in> S. (y, z) \<in> r) \<longrightarrow> (L, z) \<in> r\<rbrakk> \<Longrightarrow> L = lub S cl"
-  apply (rule lub_unique, assumption)
-   apply (simp add: isLub_def A_def r_def)
-  apply (unfold isLub_def)
-  apply (rule conjI)
-   apply (fold A_def r_def)
-   apply (rule lub_in_lattice, assumption)
-  apply (simp add: lub_upper lub_least)
-  done
+  assumes A: "S \<subseteq> A" "L \<in> A" and r: "\<forall>x \<in> S. (x, L) \<in> r" 
+     and clo: "\<And>z. \<lbrakk>z \<in> A; (\<forall>y \<in> S. (y, z) \<in> r)\<rbrakk> \<Longrightarrow> (L, z) \<in> r" 
+   shows "L = lub S cl"
+proof -
+  obtain L where "isLub S cl L"
+    using CL_imp_ex_isLub assms(1) by auto
+  then show ?thesis
+    by (simp add: antisymE A clo lub_in_lattice lub_least lub_upper r)
+qed
 
 lemma lubIa: "\<lbrakk>S \<subseteq> A; isLub S cl L\<rbrakk> \<Longrightarrow> L = lub S cl"
-  by (simp add: lubI isLub_def A_def r_def)
+  by (meson isLub_lub lub_unique)
 
 lemma isLub_in_lattice: "isLub S cl L \<Longrightarrow> L \<in> A"
   by (simp add: isLub_def  A_def)
@@ -397,22 +366,10 @@
 begin
 
 lemma glb_in_lattice: "S \<subseteq> A \<Longrightarrow> glb S cl \<in> A"
-  apply (subst glb_dual_lub)
-  apply (simp add: A_def)
-  apply (rule dualA_iff [THEN subst])
-  apply (rule CL.lub_in_lattice)
-   apply (rule dual)
-  apply (simp add: dualA_iff)
-  done
+  by (metis A_def CL.lub_in_lattice dualA_iff glb_dual_lub local.dual)
 
 lemma glb_lower: "\<lbrakk>S \<subseteq> A; x \<in> S\<rbrakk> \<Longrightarrow> (glb S cl, x) \<in> r"
-  apply (subst glb_dual_lub)
-  apply (simp add: r_def)
-  apply (rule dualr_iff [THEN subst])
-  apply (rule CL.lub_upper)
-    apply (rule dual)
-   apply (simp add: dualA_iff A_def, assumption)
-  done
+  by (metis A_def CL.lub_upper dualA_iff dualr_iff glb_dual_lub local.dual r_def)
 
 end
 
@@ -437,7 +394,12 @@
   by (simp add: A_def r_def)
 
 lemma CLF_dual: "(dual cl, f) \<in> CLF_set"
-  by (simp add: CLF_set_def  CL_dualCL monotone_dual) (simp add: dualA_iff)
+proof -
+  have "Tarski.monotone f A (order (dual cl))"
+    by (metis (no_types) A_def PO.monotone_dual PO_axioms dualA_iff monotone_f r_def)
+  then show ?thesis
+    by (simp add: A_def CLF_set_def CL_dualCL)
+qed
 
 lemma dual: "CLF (dual cl) f"
   by (rule CLF.intro) (rule CLF_dual)
@@ -462,94 +424,87 @@
 context CLF
 begin
 
-lemma lubH_le_flubH: "H = {x. (x, f x) \<in> r \<and> x \<in> A} \<Longrightarrow> (lub H cl, f (lub H cl)) \<in> r"
-  apply (rule lub_least, fast)
-   apply (rule f_in_funcset [THEN funcset_mem])
-   apply (rule lub_in_lattice, fast)
-    \<comment> \<open>\<open>\<forall>x:H. (x, f (lub H r)) \<in> r\<close>\<close>
-  apply (rule ballI)
-  apply (rule transE)
-    \<comment> \<open>instantiates \<open>(x, ???z) \<in> order cl to (x, f x)\<close>,\<close>
-    \<comment> \<open>because of the def of \<open>H\<close>\<close>
-   apply fast
-    \<comment> \<open>so it remains to show \<open>(f x, f (lub H cl)) \<in> r\<close>\<close>
-  apply (rule_tac f = "f" in monotoneE)
-     apply (rule monotone_f, fast)
-   apply (rule lub_in_lattice, fast)
-  apply (rule lub_upper, fast)
-  apply assumption
-  done
+lemma lubH_le_flubH: 
+  assumes "H = {x \<in> A. (x, f x) \<in> r}"
+  shows "(lub H cl, f (lub H cl)) \<in> r"
+proof (intro lub_least ballI)
+  show "H \<subseteq> A"
+    using assms
+    by auto
+  show "f (lub H cl) \<in> A"
+    using \<open>H \<subseteq> A\<close> f_in_funcset lub_in_lattice by auto
+  show "(x, f (lub H cl)) \<in> r" if "x \<in> H" for x
+  proof -
+    have "(f x, f (lub H cl)) \<in> r"
+      by (meson \<open>H \<subseteq> A\<close> in_mono lub_in_lattice lub_upper monotoneE monotone_f that)
+    moreover have "(x, f x) \<in> r"
+      using assms that by blast
+    ultimately show ?thesis
+      using po.transE by blast
+  qed
+qed
 
-lemma flubH_le_lubH: "\<lbrakk>H = {x. (x, f x) \<in> r \<and> x \<in> A}\<rbrakk> \<Longrightarrow> (f (lub H cl), lub H cl) \<in> r"
-  apply (rule lub_upper, fast)
-  apply (rule_tac t = "H" in ssubst, assumption)
-  apply (rule CollectI)
-  apply (rule conjI)
-   apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
-   apply (rule_tac [2] lub_in_lattice)
-   prefer 2 apply fast
-  apply (rule_tac f = f in monotoneE)
-     apply (rule monotone_f)
-    apply (blast intro: lub_in_lattice)
-   apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
-  apply (simp add: lubH_le_flubH)
-  done
+lemma flubH_le_lubH: 
+  assumes "H = {x \<in> A. (x, f x) \<in> r}"
+  shows "(f (lub H cl), lub H cl) \<in> r"
+proof -
+  have "(lub H cl, f (lub H cl)) \<in> r"
+    using assms lubH_le_flubH by blast
+  then have "(f (lub H cl), f (f (lub H cl))) \<in> r"
+    by (meson PO_imp_refl_on monotoneE monotone_f refl_on_domain)
+  then have "f (lub H cl) \<in> H"
+    by (metis (no_types, lifting) PO_imp_refl_on assms mem_Collect_eq refl_on_domain)
+  then show ?thesis
+    by (simp add: assms lub_upper)
+qed
 
-lemma lubH_is_fixp: "H = {x. (x, f x) \<in> r \<and> x \<in> A} \<Longrightarrow> lub H cl \<in> fix f A"
-  apply (simp add: fix_def)
-  apply (rule conjI)
-   apply (rule lub_in_lattice, fast)
-  apply (rule antisymE)
-   apply (simp add: flubH_le_lubH)
-  apply (simp add: lubH_le_flubH)
-  done
+
 
-lemma fix_in_H: "\<lbrakk>H = {x. (x, f x) \<in> r \<and> x \<in> A}; x \<in> P\<rbrakk> \<Longrightarrow> x \<in> H"
+lemma lubH_is_fixp: "H = {x \<in> A. (x, f x) \<in> r} \<Longrightarrow> lub H cl \<in> fix f A"
+  by (simp add: fix_def antisymE flubH_le_lubH lubH_le_flubH lub_in_lattice)
+
+lemma fix_in_H: "\<lbrakk>H = {x \<in> A. (x, f x) \<in> r}; x \<in> P\<rbrakk> \<Longrightarrow> x \<in> H"
   by (simp add: P_def fix_imp_eq [of _ f A] reflE fix_subset [of f A, THEN subsetD])
 
-lemma fixf_le_lubH: "H = {x. (x, f x) \<in> r \<and> x \<in> A} \<Longrightarrow> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
-  apply (rule ballI)
-  apply (rule lub_upper)
-   apply fast
-  apply (rule fix_in_H)
-   apply (simp_all add: P_def)
-  done
+lemma fixf_le_lubH: "H = {x \<in> A. (x, f x) \<in> r} \<Longrightarrow> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
+  by (metis (no_types, lifting) P_def fix_in_H lub_upper mem_Collect_eq subset_eq)
 
 lemma lubH_least_fixf:
-  "H = {x. (x, f x) \<in> r \<and> x \<in> A} \<Longrightarrow> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) \<longrightarrow> (lub H cl, L) \<in> r"
-  apply (rule allI)
-  apply (rule impI)
-  apply (erule bspec)
-  apply (rule lubH_is_fixp, assumption)
-  done
+  "H = {x \<in> A. (x, f x) \<in> r} \<Longrightarrow> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) \<longrightarrow> (lub H cl, L) \<in> r"
+  using lubH_is_fixp by blast
 
 
 subsection \<open>Tarski fixpoint theorem 1, first part\<close>
 
-lemma T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r \<and> x \<in> A} cl"
-  apply (rule sym)
-  apply (simp add: P_def)
-  apply (rule lubI)
-     apply (rule fix_subset)
-    apply (rule lub_in_lattice, fast)
-   apply (simp add: fixf_le_lubH)
-  apply (simp add: lubH_least_fixf)
-  done
+lemma T_thm_1_lub: "lub P cl = lub {x \<in> A. (x, f x) \<in> r} cl"
+proof -
+  have "lub {x \<in> A. (x, f x) \<in> r} cl = lub (fix f A) cl"
+  proof (rule antisymE)
+    show "(lub {x \<in> A. (x, f x) \<in> r} cl, lub (fix f A) cl) \<in> r"
+      by (simp add: fix_subset lubH_is_fixp lub_upper)
+    have "\<And>a. a \<in> fix f A \<Longrightarrow> a \<in> A"
+      by (meson fix_subset subset_iff)
+    then show "(lub (fix f A) cl, lub {x \<in> A. (x, f x) \<in> r} cl) \<in> r"
+      by (simp add: fix_subset fixf_le_lubH lubH_is_fixp lub_least)
+  qed
+  then show ?thesis
+    using P_def by auto
+qed
 
-lemma glbH_is_fixp: "H = {x. (f x, x) \<in> r \<and> x \<in> A} \<Longrightarrow> glb H cl \<in> P"
+lemma glbH_is_fixp: 
+  assumes "H = {x \<in> A. (f x, x) \<in> r}" shows "glb H cl \<in> P"
   \<comment> \<open>Tarski for glb\<close>
-  apply (simp add: glb_dual_lub P_def A_def r_def)
-  apply (rule dualA_iff [THEN subst])
-  apply (rule CLF.lubH_is_fixp)
-   apply (rule dual)
-  apply (simp add: dualr_iff dualA_iff)
-  done
+proof -
+  have "glb H cl \<in> fix f (pset (dual cl))"
+    using assms CLF.lubH_is_fixp [OF dual] PO.dualr_iff PO_axioms  
+    by (fastforce simp add: A_def r_def glb_dual_lub)
+  then show ?thesis
+    by (simp add: A_def P_def)
+qed
 
-lemma T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r \<and> x \<in> A} cl"
-  apply (simp add: glb_dual_lub P_def A_def r_def)
-  apply (rule dualA_iff [THEN subst])
-  apply (simp add: CLF.T_thm_1_lub [of _ f, OF dual] dualPO CL_dualCL CLF_dual dualr_iff)
-  done
+lemma T_thm_1_glb: "glb P cl = glb {x \<in> A. (f x, x) \<in> r} cl"
+  unfolding glb_dual_lub P_def A_def r_def
+  using CLF.T_thm_1_lub dualA_iff dualr_iff local.dual by force
 
 
 subsection \<open>interval\<close>
@@ -572,31 +527,28 @@
 lemma a_less_lub: "\<lbrakk>S \<subseteq> A; S \<noteq> {}; \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r\<rbrakk> \<Longrightarrow> (a, L) \<in> r"
   by (blast intro: transE)
 
-lemma glb_less_b: "\<lbrakk>S \<subseteq> A; S \<noteq> {}; \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r\<rbrakk> \<Longrightarrow> (G, b) \<in> r"
-  by (blast intro: transE)
-
 lemma S_intv_cl: "\<lbrakk>a \<in> A; b \<in> A; S \<subseteq> interval r a b\<rbrakk> \<Longrightarrow> S \<subseteq> A"
   by (simp add: subset_trans [OF _ interval_subset])
 
 lemma L_in_interval:
-  "\<lbrakk>a \<in> A; b \<in> A; S \<subseteq> interval r a b;
-    S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {}\<rbrakk> \<Longrightarrow> L \<in> interval r a b"
-  apply (rule intervalI)
-   apply (rule a_less_lub)
-      prefer 2 apply assumption
-     apply (simp add: S_intv_cl)
-    apply (rule ballI)
-    apply (simp add: interval_lemma1)
-   apply (simp add: isLub_upper)
-    \<comment> \<open>\<open>(L, b) \<in> r\<close>\<close>
-  apply (simp add: isLub_least interval_lemma2)
-  done
+  assumes "b \<in> A" and S: "S \<subseteq> interval r a b" "isLub S cl L" "S \<noteq> {}"
+  shows "L \<in> interval r a b"
+proof (rule intervalI)
+  show "(a, L) \<in> r"
+    by (meson PO_imp_trans all_not_in_conv S interval_lemma1 isLub_upper transD)
+  show "(L, b) \<in> r"
+    using \<open>b \<in> A\<close> assms interval_lemma2 isLub_least by auto
+qed
 
 lemma G_in_interval:
-  "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G; S \<noteq> {}\<rbrakk>
-    \<Longrightarrow> G \<in> interval r a b"
-  by (simp add: interval_dual)
-    (simp add: CLF.L_in_interval [of _ f, OF dual] dualA_iff A_def isGlb_dual_isLub)
+  assumes "b \<in> A" and S: "S \<subseteq> interval r a b" "isGlb S cl G" "S \<noteq> {}"
+  shows "G \<in> interval r a b"
+proof -
+  have "a \<in> A"
+    using S(1) \<open>S \<noteq> {}\<close> interval_lemma1 rel_imp_elem by blast
+  with assms show ?thesis
+    by (metis (no_types) A_def CLF.L_in_interval dualA_iff interval_dual isGlb_dual_isLub local.dual)
+qed
 
 lemma intervalPO:
   "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}\<rbrakk>
@@ -604,66 +556,38 @@
   by (rule po_subset_po) (simp add: interval_subset)
 
 lemma intv_CL_lub:
-  "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}\<rbrakk> \<Longrightarrow>
-    \<forall>S. S \<subseteq> interval r a b \<longrightarrow>
-      (\<exists>L. isLub S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr>  L)"
-  apply (intro strip)
-  apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
-    prefer 2 apply assumption
-   apply assumption
-  apply (erule exE)
-    \<comment> \<open>define the lub for the interval as\<close>
-  apply (rule_tac x = "if S = {} then a else L" in exI)
-  apply (simp (no_asm_simp) add: isLub_def split del: if_split)
-  apply (intro impI conjI)
-    \<comment> \<open>\<open>(if S = {} then a else L) \<in> interval r a b\<close>\<close>
-    apply (simp add: CL_imp_PO L_in_interval)
-    apply (simp add: left_in_interval)
-    \<comment> \<open>lub prop 1\<close>
-   apply (case_tac "S = {}")
-    \<comment> \<open>\<open>S = {}, y \<in> S = False \<Longrightarrow> everything\<close>\<close>
-    apply fast
-    \<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
-   apply simp
-    \<comment> \<open>\<open>\<forall>y\<in>S. (y, L) \<in> induced (interval r a b) r\<close>\<close>
-   apply (rule ballI)
-   apply (simp add: induced_def  L_in_interval)
-   apply (rule conjI)
-    apply (rule subsetD)
-     apply (simp add: S_intv_cl, assumption)
-   apply (simp add: isLub_upper)
-    \<comment> \<open>\<open>\<forall>z\<in>interval r a b.
-        (\<forall>y\<in>S. (y, z) \<in> induced (interval r a b) r \<longrightarrow>
-        (if S = {} then a else L, z) \<in> induced (interval r a b) r\<close>\<close>
-  apply (rule ballI)
-  apply (rule impI)
-  apply (case_tac "S = {}")
-    \<comment> \<open>\<open>S = {}\<close>\<close>
-   apply simp
-   apply (simp add: induced_def  interval_def)
-   apply (rule conjI)
-    apply (rule reflE, assumption)
-   apply (rule interval_not_empty)
-   apply (simp add: interval_def)
-    \<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
-  apply simp
-  apply (simp add: induced_def  L_in_interval)
-  apply (rule isLub_least, assumption)
-   apply (rule subsetD)
-    prefer 2 apply assumption
-   apply (simp add: S_intv_cl, fast)
-  done
+  assumes "a \<in> A" "b \<in> A" "interval r a b \<noteq> {}" and S: "S \<subseteq> interval r a b"
+  shows "\<exists>L. isLub S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr>  L"
+proof -
+  obtain L where L: "isLub S cl L"
+    by (meson CL_imp_ex_isLub S_intv_cl assms(1) assms(2) assms(4))
+  show ?thesis
+    unfolding isLub_def potype.simps
+    proof (intro exI impI conjI ballI)
+    let ?L = "(if S = {} then a else L)"
+    show Lin: "?L \<in> interval r a b"
+      using L L_in_interval assms left_in_interval by auto
+    show "(y, ?L) \<in> induced (interval r a b) r" if "y \<in> S" for y
+    proof -
+      have "S \<noteq> {}"
+        using that by blast
+      then show ?thesis
+        using L Lin S indI isLub_upper that by auto
+    qed
+  show "(?L, z) \<in> induced (interval r a b) r"
+    if "z \<in> interval r a b" and "\<forall>y\<in>S. (y, z) \<in> induced (interval r a b) r" for z
+      using that L
+      apply (simp add: isLub_def induced_def interval_imp_mem)
+      by (metis (full_types) A_def Lin \<open>a \<in> A\<close> \<open>b \<in> A\<close> interval_subset r_def subset_eq)
+  qed
+qed
 
 lemmas intv_CL_glb = intv_CL_lub [THEN Rdual]
 
 lemma interval_is_sublattice: "\<lbrakk>a \<in> A; b \<in> A; interval r a b \<noteq> {}\<rbrakk> \<Longrightarrow> interval r a b <<= cl"
   apply (rule sublatticeI)
    apply (simp add: interval_subset)
-  apply (rule CompleteLatticeI)
-    apply (simp add: intervalPO)
-   apply (simp add: intv_CL_lub)
-  apply (simp add: intv_CL_glb)
-  done
+  by (simp add: CompleteLatticeI intervalPO intv_CL_glb intv_CL_lub)
 
 lemmas interv_is_compl_latt = interval_is_sublattice [THEN sublattice_imp_CL]
 
@@ -671,55 +595,32 @@
 subsection \<open>Top and Bottom\<close>
 
 lemma Top_dual_Bot: "Top cl = Bot (dual cl)"
-  by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
+  by (simp add: Top_def Bot_def least_def greatest_def)
 
 lemma Bot_dual_Top: "Bot cl = Top (dual cl)"
-  by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
+  by (simp add: Top_def Bot_def least_def greatest_def)
 
 lemma Bot_in_lattice: "Bot cl \<in> A"
-  apply (simp add: Bot_def least_def)
+  unfolding Bot_def least_def
   apply (rule_tac a = "glb A cl" in someI2)
-   apply (simp_all add: glb_in_lattice glb_lower r_def [symmetric] A_def [symmetric])
-  done
+  using glb_in_lattice glb_lower by (auto simp: A_def r_def)
 
 lemma Top_in_lattice: "Top cl \<in> A"
-  apply (simp add: Top_dual_Bot A_def)
-  apply (rule dualA_iff [THEN subst])
-  apply (rule CLF.Bot_in_lattice [OF dual])
-  done
+  using A_def CLF.Bot_in_lattice Top_dual_Bot local.dual by force
 
 lemma Top_prop: "x \<in> A \<Longrightarrow> (x, Top cl) \<in> r"
-  apply (simp add: Top_def greatest_def)
+  unfolding Top_def greatest_def
   apply (rule_tac a = "lub A cl" in someI2)
-   apply (rule someI2)
-    apply (simp_all add: lub_in_lattice lub_upper
-      r_def [symmetric] A_def [symmetric])
-  done
+  using lub_in_lattice lub_upper by (auto simp: A_def r_def)
 
 lemma Bot_prop: "x \<in> A \<Longrightarrow> (Bot cl, x) \<in> r"
-  apply (simp add: Bot_dual_Top r_def)
-  apply (rule dualr_iff [THEN subst])
-  apply (rule CLF.Top_prop [OF dual])
-  apply (simp add: dualA_iff A_def)
-  done
+  using A_def Bot_dual_Top CLF.Top_prop dualA_iff dualr_iff local.dual r_def by fastforce
 
 lemma Top_intv_not_empty: "x \<in> A \<Longrightarrow> interval r x (Top cl) \<noteq> {}"
-  apply (rule notI)
-  apply (drule_tac a = "Top cl" in equals0D)
-  apply (simp add: interval_def)
-  apply (simp add: refl_on_def Top_in_lattice Top_prop)
-  done
+  using Top_prop intervalI reflE by force
 
 lemma Bot_intv_not_empty: "x \<in> A \<Longrightarrow> interval r (Bot cl) x \<noteq> {}"
-  apply (simp add: Bot_dual_Top)
-  apply (subst interval_dual)
-    prefer 2 apply assumption
-   apply (simp add: A_def)
-   apply (rule dualA_iff [THEN subst])
-   apply (rule CLF.Top_in_lattice [OF dual])
-  apply (rule CLF.Top_intv_not_empty [OF dual])
-  apply (simp add: dualA_iff A_def)
-  done
+  using Bot_dual_Top Bot_prop intervalI reflE by fastforce
 
 
 subsection \<open>fixed points form a partial order\<close>
@@ -739,47 +640,34 @@
   by (rule Y_subset_A [THEN lub_in_lattice])
 
 lemma lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
-  apply (rule lub_least)
-    apply (rule Y_subset_A)
-   apply (rule f_in_funcset [THEN funcset_mem])
-   apply (rule lubY_in_A)
-    \<comment> \<open>\<open>Y \<subseteq> P \<Longrightarrow> f x = x\<close>\<close>
-  apply (rule ballI)
-  apply (rule_tac t = x in fix_imp_eq [THEN subst])
-   apply (erule Y_ss [simplified P_def, THEN subsetD])
-    \<comment> \<open>\<open>reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r\<close> by monotonicity\<close>
-  apply (rule_tac f = "f" in monotoneE)
-     apply (rule monotone_f)
-    apply (simp add: Y_subset_A [THEN subsetD])
-   apply (rule lubY_in_A)
-  apply (simp add: lub_upper Y_subset_A)
-  done
+proof (intro lub_least Y_subset_A ballI)
+  show "f (lub Y cl) \<in> A"
+    by (meson Tarski.monotone_def lubY_in_A monotone_f reflE rel_imp_elem)
+  show "(x, f (lub Y cl)) \<in> r" if "x \<in> Y" for x
+  proof 
+    have "\<And>A. Y \<subseteq> A \<Longrightarrow> x \<in> A"
+      using that by blast
+    moreover have "(x, lub Y cl) \<in> r"
+      using that by (simp add: Y_subset_A lub_upper)
+    ultimately show "(x, f (lub Y cl)) \<in> r"
+      by (metis (no_types) Tarski.Y_ss Tarski_axioms Y_subset_A fix_imp_eq lubY_in_A monotoneE monotone_f)
+  qed auto
+qed
 
 lemma intY1_subset: "intY1 \<subseteq> A"
-  apply (unfold intY1_def)
-  apply (rule interval_subset)
-   apply (rule lubY_in_A)
-  apply (rule Top_in_lattice)
-  done
+  unfolding intY1_def using Top_in_lattice interval_subset lubY_in_A by auto
 
 lemmas intY1_elem = intY1_subset [THEN subsetD]
 
-lemma intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
-  apply (simp add: intY1_def  interval_def)
-  apply (rule conjI)
-   apply (rule transE)
-    apply (rule lubY_le_flubY)
-    \<comment> \<open>\<open>(f (lub Y cl), f x) \<in> r\<close>\<close>
-   apply (rule_tac f=f in monotoneE)
-      apply (rule monotone_f)
-     apply (rule lubY_in_A)
-    apply (simp add: intY1_def interval_def  intY1_elem)
-   apply (simp add: intY1_def  interval_def)
-    \<comment> \<open>\<open>(f x, Top cl) \<in> r\<close>\<close>
-  apply (rule Top_prop)
-  apply (rule f_in_funcset [THEN funcset_mem])
-  apply (simp add: intY1_def interval_def  intY1_elem)
-  done
+lemma intY1_f_closed:   
+  assumes "x \<in> intY1" shows "f x \<in> intY1"
+proof (simp add: intY1_def interval_def, rule conjI)
+  show "(lub Y cl, f x) \<in> r"
+    using assms intY1_elem interval_imp_mem lubY_in_A unfolding intY1_def
+    using lubY_le_flubY monotoneE monotone_f po.transE by blast
+  then show "(f x, Top cl) \<in> r"
+    by (meson PO_imp_refl_on Top_prop refl_onD2)
+qed
 
 lemma intY1_mono: "monotone (\<lambda> x \<in> intY1. f x) intY1 (induced intY1 r)"
   apply (auto simp add: monotone_def induced_def intY1_f_closed)
@@ -787,92 +675,68 @@
   done
 
 lemma intY1_is_cl: "\<lparr>pset = intY1, order = induced intY1 r\<rparr> \<in> CompleteLattice"
-  apply (unfold intY1_def)
-  apply (rule interv_is_compl_latt)
-    apply (rule lubY_in_A)
-   apply (rule Top_in_lattice)
-  apply (rule Top_intv_not_empty)
-  apply (rule lubY_in_A)
-  done
+  unfolding intY1_def
+  by (simp add: Top_in_lattice Top_intv_not_empty interv_is_compl_latt lubY_in_A)
 
 lemma v_in_P: "v \<in> P"
-  apply (unfold P_def)
-  apply (rule_tac A = intY1 in fixf_subset)
-   apply (rule intY1_subset)
-  unfolding v_def
-  apply (rule CLF.glbH_is_fixp
-      [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])
-   apply auto
-    apply (rule intY1_is_cl)
-   apply (erule intY1_f_closed)
-  apply (rule intY1_mono)
-  done
+proof -
+  have "v \<in> fix (restrict f intY1) intY1"
+    unfolding v_def
+    apply (rule CLF.glbH_is_fixp
+        [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])
+    using intY1_f_closed intY1_is_cl intY1_mono apply blast+
+    done
+  then show ?thesis
+    unfolding P_def
+  by (meson fixf_subset intY1_subset)
+qed
 
 lemma z_in_interval: "\<lbrakk>z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r\<rbrakk> \<Longrightarrow> z \<in> intY1"
-  apply (unfold intY1_def P_def)
-  apply (rule intervalI)
-   prefer 2
-   apply (erule fix_subset [THEN subsetD, THEN Top_prop])
-  apply (rule lub_least)
-    apply (rule Y_subset_A)
-   apply (fast elim!: fix_subset [THEN subsetD])
-  apply (simp add: induced_def)
-  done
-
-lemma f'z_in_int_rel: "\<lbrakk>z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r\<rbrakk>
-  \<Longrightarrow> ((\<lambda>x \<in> intY1. f x) z, z) \<in> induced intY1 r"
-  by (simp add: induced_def  intY1_f_closed z_in_interval P_def)
-    (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD] reflE)
+  unfolding intY1_def P_def
+  by (meson Top_prop Y_subset_A fix_subset in_mono indE intervalI lub_least)
 
 lemma tarski_full_lemma: "\<exists>L. isLub Y \<lparr>pset = P, order = induced P r\<rparr> L"
-  apply (rule_tac x = "v" in exI)
-  apply (simp add: isLub_def)
-    \<comment> \<open>\<open>v \<in> P\<close>\<close>
-  apply (simp add: v_in_P)
-  apply (rule conjI)
-    \<comment> \<open>\<open>v\<close> is lub\<close>
-    \<comment> \<open>\<open>1. \<forall>y:Y. (y, v) \<in> induced P r\<close>\<close>
-   apply (rule ballI)
-   apply (simp add: induced_def subsetD v_in_P)
-   apply (rule conjI)
-    apply (erule Y_ss [THEN subsetD])
-   apply (rule_tac b = "lub Y cl" in transE)
-    apply (rule lub_upper)
-     apply (rule Y_subset_A, assumption)
-   apply (rule_tac b = "Top cl" in interval_imp_mem)
-   apply (simp add: v_def)
-   apply (fold intY1_def)
-   apply (rule CL.glb_in_lattice [OF CL.intro [OF intY1_is_cl], simplified])
-   apply auto
-  apply (rule indI)
-    prefer 3 apply assumption
-   prefer 2 apply (simp add: v_in_P)
-  apply (unfold v_def)
-  apply (rule indE)
-   apply (rule_tac [2] intY1_subset)
-  apply (rule CL.glb_lower [OF CL.intro [OF intY1_is_cl], simplified])
-   apply (simp add: CL_imp_PO intY1_is_cl)
-   apply force
-  apply (simp add: induced_def intY1_f_closed z_in_interval)
-  apply (simp add: P_def fix_imp_eq [of _ f A] reflE fix_subset [of f A, THEN subsetD])
-  done
+proof
+  have "(y, v) \<in> induced P r" if "y \<in> Y" for y
+  proof -
+    have "(y, lub Y cl) \<in> r"
+      by (simp add: Y_subset_A lub_upper that)
+    moreover have "(lub Y cl, v) \<in> r"
+      by (metis (no_types, lifting) CL.glb_in_lattice CL.intro intY1_def intY1_is_cl interval_imp_mem lub_dual_glb mem_Collect_eq select_convs(1) subsetI v_def)
+    ultimately have "(y, v) \<in> r"
+      using po.transE by blast
+    then show ?thesis
+      using Y_ss indI that v_in_P by auto
+  qed
+  moreover have "(v, z) \<in> induced P r" if "z \<in> P" "\<forall>y\<in>Y. (y, z) \<in> induced P r" for z
+  proof (rule indI)
+    have "((\<lambda>x \<in> intY1. f x) z, z) \<in> induced intY1 r"
+      by (metis P_def fix_imp_eq in_mono indI intY1_subset reflE restrict_apply' that z_in_interval)
+    then show "(v, z) \<in> r"
+      by (metis (no_types, lifting) CL.glb_lower CL_def indE intY1_is_cl intY1_subset mem_Collect_eq select_convs(1,2) subsetI that v_def z_in_interval)
+  qed (auto simp: that v_in_P)
+  ultimately
+  show "isLub Y \<lparr>pset = P, order = induced P r\<rparr> v"
+    by (simp add: isLub_def v_in_P)
+qed
 
 end
 
 lemma CompleteLatticeI_simp:
-  "\<lbrakk>\<lparr>pset = A, order = r\<rparr> \<in> PartialOrder;
-     \<forall>S. S \<subseteq> A \<longrightarrow> (\<exists>L. isLub S \<lparr>pset = A, order = r\<rparr>  L)\<rbrakk>
-    \<Longrightarrow> \<lparr>pset = A, order = r\<rparr> \<in> CompleteLattice"
-  by (simp add: CompleteLatticeI Rdual)
+  "\<lbrakk>po \<in> PartialOrder; \<And>S. S \<subseteq> A \<Longrightarrow> \<exists>L. isLub S po  L; A = pset po\<rbrakk>
+    \<Longrightarrow> po \<in> CompleteLattice"
+  by (metis CompleteLatticeI Rdual)
+
 
 theorem (in CLF) Tarski_full: "\<lparr>pset = P, order = induced P r\<rparr> \<in> CompleteLattice"
-  apply (rule CompleteLatticeI_simp)
-   apply (rule fixf_po)
-  apply clarify
-  apply (simp add: P_def A_def r_def)
-  apply (rule Tarski.tarski_full_lemma [OF Tarski.intro [OF _ Tarski_axioms.intro]])
-proof -
-  show "CLF cl f" ..
-qed
+proof (intro CompleteLatticeI_simp allI impI)
+  show "\<lparr>pset = P, order = induced P r\<rparr> \<in> PartialOrder"
+    by (simp add: fixf_po)
+  show "\<And>S. S \<subseteq> P \<Longrightarrow> \<exists>L. isLub S \<lparr>pset = P, order = induced P r\<rparr> L"
+    unfolding P_def A_def r_def
+  proof (rule Tarski.tarski_full_lemma [OF Tarski.intro [OF _ Tarski_axioms.intro]])
+    show "CLF cl f" ..
+  qed
+qed auto
 
 end