src/HOL/Library/Product_Order.thy
changeset 63561 fba08009ff3e
parent 62343 24106dc44def
child 63972 c98d1dd7eba1
equal deleted inserted replaced
63559:113cee845044 63561:fba08009ff3e
     3 *)
     3 *)
     4 
     4 
     5 section \<open>Pointwise order on product types\<close>
     5 section \<open>Pointwise order on product types\<close>
     6 
     6 
     7 theory Product_Order
     7 theory Product_Order
     8 imports Product_plus Conditionally_Complete_Lattices
     8 imports Product_plus
     9 begin
     9 begin
    10 
    10 
    11 subsection \<open>Pointwise ordering\<close>
    11 subsection \<open>Pointwise ordering\<close>
    12 
    12 
    13 instantiation prod :: (ord, ord) ord
    13 instantiation prod :: (ord, ord) ord
   241   case 2
   241   case 2
   242   then show ?case
   242   then show ?case
   243     by (auto simp: inf_prod_def Sup_prod_def SUP_prod_alt_def inf_Sup inf_SUP comp_def)
   243     by (auto simp: inf_prod_def Sup_prod_def SUP_prod_alt_def inf_Sup inf_SUP comp_def)
   244 qed
   244 qed
   245 
   245 
   246 end
   246 subsection \<open>Bekic's Theorem\<close>
   247 
   247 text \<open>
       
   248   Simultaneous fixed points over pairs can be written in terms of separate fixed points.
       
   249   Transliterated from HOLCF.Fix by Peter Gammie
       
   250 \<close>
       
   251 
       
   252 lemma lfp_prod:
       
   253   fixes F :: "'a::complete_lattice \<times> 'b::complete_lattice \<Rightarrow> 'a \<times> 'b"
       
   254   assumes "mono F"
       
   255   shows "lfp F = (lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))),
       
   256                  (lfp (\<lambda>y. snd (F (lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))), y)))))"
       
   257   (is "lfp F = (?x, ?y)")
       
   258 proof(rule lfp_eqI[OF assms])
       
   259   have 1: "fst (F (?x, ?y)) = ?x"
       
   260     by (rule trans [symmetric, OF lfp_unfold])
       
   261        (blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono lfp_mono)+
       
   262   have 2: "snd (F (?x, ?y)) = ?y"
       
   263     by (rule trans [symmetric, OF lfp_unfold])
       
   264        (blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono lfp_mono)+
       
   265   from 1 2 show "F (?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)
       
   266 next
       
   267   fix z assume F_z: "F z = z"
       
   268   obtain x y where z: "z = (x, y)" by (rule prod.exhaust)
       
   269   from F_z z have F_x: "fst (F (x, y)) = x" by simp
       
   270   from F_z z have F_y: "snd (F (x, y)) = y" by simp
       
   271   let ?y1 = "lfp (\<lambda>y. snd (F (x, y)))"
       
   272   have "?y1 \<le> y" by (rule lfp_lowerbound, simp add: F_y)
       
   273   hence "fst (F (x, ?y1)) \<le> fst (F (x, y))"
       
   274     by (simp add: assms fst_mono monoD)
       
   275   hence "fst (F (x, ?y1)) \<le> x" using F_x by simp
       
   276   hence 1: "?x \<le> x" by (simp add: lfp_lowerbound)
       
   277   hence "snd (F (?x, y)) \<le> snd (F (x, y))"
       
   278     by (simp add: assms snd_mono monoD)
       
   279   hence "snd (F (?x, y)) \<le> y" using F_y by simp
       
   280   hence 2: "?y \<le> y" by (simp add: lfp_lowerbound)
       
   281   show "(?x, ?y) \<le> z" using z 1 2 by simp
       
   282 qed
       
   283 
       
   284 lemma gfp_prod:
       
   285   fixes F :: "'a::complete_lattice \<times> 'b::complete_lattice \<Rightarrow> 'a \<times> 'b"
       
   286   assumes "mono F"
       
   287   shows "gfp F = (gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y)))))),
       
   288                  (gfp (\<lambda>y. snd (F (gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y)))))), y)))))"
       
   289   (is "gfp F = (?x, ?y)")
       
   290 proof(rule gfp_eqI[OF assms])
       
   291   have 1: "fst (F (?x, ?y)) = ?x"
       
   292     by (rule trans [symmetric, OF gfp_unfold])
       
   293        (blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono gfp_mono)+
       
   294   have 2: "snd (F (?x, ?y)) = ?y"
       
   295     by (rule trans [symmetric, OF gfp_unfold])
       
   296        (blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono gfp_mono)+
       
   297   from 1 2 show "F (?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)
       
   298 next
       
   299   fix z assume F_z: "F z = z"
       
   300   obtain x y where z: "z = (x, y)" by (rule prod.exhaust)
       
   301   from F_z z have F_x: "fst (F (x, y)) = x" by simp
       
   302   from F_z z have F_y: "snd (F (x, y)) = y" by simp
       
   303   let ?y1 = "gfp (\<lambda>y. snd (F (x, y)))"
       
   304   have "y \<le> ?y1" by (rule gfp_upperbound, simp add: F_y)
       
   305   hence "fst (F (x, y)) \<le> fst (F (x, ?y1))"
       
   306     by (simp add: assms fst_mono monoD)
       
   307   hence "x \<le> fst (F (x, ?y1))" using F_x by simp
       
   308   hence 1: "x \<le> ?x" by (simp add: gfp_upperbound)
       
   309   hence "snd (F (x, y)) \<le> snd (F (?x, y))"
       
   310     by (simp add: assms snd_mono monoD)
       
   311   hence "y \<le> snd (F (?x, y))" using F_y by simp
       
   312   hence 2: "y \<le> ?y" by (simp add: gfp_upperbound)
       
   313   show "z \<le> (?x, ?y)" using z 1 2 by simp
       
   314 qed
       
   315 
       
   316 end