Theory Product_Order

(*  Title:      HOL/Library/Product_Order.thy
Author:     Brian Huffman
*)

section Pointwise order on product types

theory Product_Order
imports Product_Plus
begin

subsection Pointwise ordering

instantiation prod :: (ord, ord) ord
begin

definition
"x  y  fst x  fst y  snd x  snd y"

definition
"(x::'a × 'b) < y  x  y  ¬ y  x"

instance ..

end

lemma fst_mono: "x  y  fst x  fst y"
unfolding less_eq_prod_def by simp

lemma snd_mono: "x  y  snd x  snd y"
unfolding less_eq_prod_def by simp

lemma Pair_mono: "x  x'  y  y'  (x, y)  (x', y')"
unfolding less_eq_prod_def by simp

lemma Pair_le [simp]: "(a, b)  (c, d)  a  c  b  d"
unfolding less_eq_prod_def by simp

instance prod :: (preorder, preorder) preorder
proof
fix x y z :: "'a × 'b"
show "x < y  x  y  ¬ y  x"
by (rule less_prod_def)
show "x  x"
unfolding less_eq_prod_def
by fast
assume "x  y" and "y  z" thus "x  z"
unfolding less_eq_prod_def
by (fast elim: order_trans)
qed

instance prod :: (order, order) order
by standard auto

subsection Binary infimum and supremum

instantiation prod :: (inf, inf) inf
begin

definition "inf x y = (inf (fst x) (fst y), inf (snd x) (snd y))"

lemma inf_Pair_Pair [simp]: "inf (a, b) (c, d) = (inf a c, inf b d)"
unfolding inf_prod_def by simp

lemma fst_inf [simp]: "fst (inf x y) = inf (fst x) (fst y)"
unfolding inf_prod_def by simp

lemma snd_inf [simp]: "snd (inf x y) = inf (snd x) (snd y)"
unfolding inf_prod_def by simp

instance ..

end

instance prod :: (semilattice_inf, semilattice_inf) semilattice_inf
by standard auto

instantiation prod :: (sup, sup) sup
begin

definition
"sup x y = (sup (fst x) (fst y), sup (snd x) (snd y))"

lemma sup_Pair_Pair [simp]: "sup (a, b) (c, d) = (sup a c, sup b d)"
unfolding sup_prod_def by simp

lemma fst_sup [simp]: "fst (sup x y) = sup (fst x) (fst y)"
unfolding sup_prod_def by simp

lemma snd_sup [simp]: "snd (sup x y) = sup (snd x) (snd y)"
unfolding sup_prod_def by simp

instance ..

end

instance prod :: (semilattice_sup, semilattice_sup) semilattice_sup
by standard auto

instance prod :: (lattice, lattice) lattice ..

instance prod :: (distrib_lattice, distrib_lattice) distrib_lattice
by standard (auto simp add: sup_inf_distrib1)

subsection Top and bottom elements

instantiation prod :: (top, top) top
begin

definition
"top = (top, top)"

instance ..

end

lemma fst_top [simp]:
unfolding top_prod_def by simp

lemma snd_top [simp]:
unfolding top_prod_def by simp

lemma Pair_top_top: "(top, top) = top"
unfolding top_prod_def by simp

instance prod :: (order_top, order_top) order_top
by standard (auto simp add: top_prod_def)

instantiation prod :: (bot, bot) bot
begin

definition
"bot = (bot, bot)"

instance ..

end

lemma fst_bot [simp]:
unfolding bot_prod_def by simp

lemma snd_bot [simp]:
unfolding bot_prod_def by simp

lemma Pair_bot_bot: "(bot, bot) = bot"
unfolding bot_prod_def by simp

instance prod :: (order_bot, order_bot) order_bot
by standard (auto simp add: bot_prod_def)

instance prod :: (bounded_lattice, bounded_lattice) bounded_lattice ..

instance prod :: (boolean_algebra, boolean_algebra) boolean_algebra
by standard (auto simp add: prod_eqI diff_eq)

subsection Complete lattice operations

instantiation prod :: (Inf, Inf) Inf
begin

definition "Inf A = (INF xA. fst x, INF xA. snd x)"

instance ..

end

instantiation prod :: (Sup, Sup) Sup
begin

definition "Sup A = (SUP xA. fst x, SUP xA. snd x)"

instance ..

end

instance prod :: (conditionally_complete_lattice, conditionally_complete_lattice)
conditionally_complete_lattice
by standard (force simp: less_eq_prod_def Inf_prod_def Sup_prod_def bdd_below_def bdd_above_def
intro!: cInf_lower cSup_upper cInf_greatest cSup_least)+

instance prod :: (complete_lattice, complete_lattice) complete_lattice
by standard (simp_all add: less_eq_prod_def Inf_prod_def Sup_prod_def
INF_lower SUP_upper le_INF_iff SUP_le_iff bot_prod_def top_prod_def)

lemma fst_Inf: "fst (Inf A) = (INF xA. fst x)"

lemma fst_INF: "fst (INF xA. f x) = (INF xA. fst (f x))"

lemma fst_Sup: "fst (Sup A) = (SUP xA. fst x)"

lemma fst_SUP: "fst (SUP xA. f x) = (SUP xA. fst (f x))"

lemma snd_Inf: "snd (Inf A) = (INF xA. snd x)"

lemma snd_INF: "snd (INF xA. f x) = (INF xA. snd (f x))"

lemma snd_Sup: "snd (Sup A) = (SUP xA. snd x)"

lemma snd_SUP: "snd (SUP xA. f x) = (SUP xA. snd (f x))"

lemma INF_Pair: "(INF xA. (f x, g x)) = (INF xA. f x, INF xA. g x)"

lemma SUP_Pair: "(SUP xA. (f x, g x)) = (SUP xA. f x, SUP xA. g x)"

text Alternative formulations for set infima and suprema over the product
of two complete lattices:

lemma INF_prod_alt_def: contributor Alessandro Coglio
"Inf (f ` A) = (Inf ((fst  f) ` A), Inf ((snd  f) ` A))"

lemma SUP_prod_alt_def: contributor Alessandro Coglio
"Sup (f ` A) = (Sup ((fst  f) ` A), Sup((snd  f) ` A))"

subsection Complete distributive lattices

instance prod :: (complete_distrib_lattice, complete_distrib_lattice) complete_distrib_lattice contributor Alessandro Coglio
proof
fix A::"('a×'b) set set"
show "Inf (Sup ` A)  Sup (Inf ` {f ` A |f. YA. f Y  Y})"
by (simp add: Inf_prod_def Sup_prod_def INF_SUP_set image_image)
qed

subsection Bekic's Theorem
text
Simultaneous fixed points over pairs can be written in terms of separate fixed points.
Transliterated from HOLCF.Fix by Peter Gammie

lemma lfp_prod:
fixes F :: "'a::complete_lattice × 'b::complete_lattice  'a × 'b"
assumes "mono F"
shows "lfp F = (lfp (λx. fst (F (x, lfp (λy. snd (F (x, y)))))),
(lfp (λy. snd (F (lfp (λx. fst (F (x, lfp (λy. snd (F (x, y)))))), y)))))"
(is "lfp F = (?x, ?y)")
proof(rule lfp_eqI[OF assms])
have 1: "fst (F (?x, ?y)) = ?x"
by (rule trans [symmetric, OF lfp_unfold])
(blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono lfp_mono)+
have 2: "snd (F (?x, ?y)) = ?y"
by (rule trans [symmetric, OF lfp_unfold])
(blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono lfp_mono)+
from 1 2 show "F (?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)
next
fix z assume F_z: "F z = z"
obtain x y where z: "z = (x, y)" by (rule prod.exhaust)
from F_z z have F_x: "fst (F (x, y)) = x" by simp
from F_z z have F_y: "snd (F (x, y)) = y" by simp
let ?y1 = "lfp (λy. snd (F (x, y)))"
have "?y1  y" by (rule lfp_lowerbound, simp add: F_y)
hence "fst (F (x, ?y1))  fst (F (x, y))"
by (simp add: assms fst_mono monoD)
hence "fst (F (x, ?y1))  x" using F_x by simp
hence 1: "?x  x" by (simp add: lfp_lowerbound)
hence "snd (F (?x, y))  snd (F (x, y))"
by (simp add: assms snd_mono monoD)
hence "snd (F (?x, y))  y" using F_y by simp
hence 2: "?y  y" by (simp add: lfp_lowerbound)
show "(?x, ?y)  z" using z 1 2 by simp
qed

lemma gfp_prod:
fixes F :: "'a::complete_lattice × 'b::complete_lattice  'a × 'b"
assumes "mono F"
shows "gfp F = (gfp (λx. fst (F (x, gfp (λy. snd (F (x, y)))))),
(gfp (λy. snd (F (gfp (λx. fst (F (x, gfp (λy. snd (F (x, y)))))), y)))))"
(is "gfp F = (?x, ?y)")
proof(rule gfp_eqI[OF assms])
have 1: "fst (F (?x, ?y)) = ?x"
by (rule trans [symmetric, OF gfp_unfold])
(blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono gfp_mono)+
have 2: "snd (F (?x, ?y)) = ?y"
by (rule trans [symmetric, OF gfp_unfold])
(blast intro!: monoI monoD[OF assms(1)] fst_mono snd_mono Pair_mono gfp_mono)+
from 1 2 show "F (?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)
next
fix z assume F_z: "F z = z"
obtain x y where z: "z = (x, y)" by (rule prod.exhaust)
from F_z z have F_x: "fst (F (x, y)) = x" by simp
from F_z z have F_y: "snd (F (x, y)) = y" by simp
let ?y1 = "gfp (λy. snd (F (x, y)))"
have "y  ?y1" by (rule gfp_upperbound, simp add: F_y)
hence "fst (F (x, y))  fst (F (x, ?y1))"
by (simp add: assms fst_mono monoD)
hence "x  fst (F (x, ?y1))" using F_x by simp
hence 1: "x  ?x" by (simp add: gfp_upperbound)
hence "snd (F (x, y))  snd (F (?x, y))"
by (simp add: assms snd_mono monoD)
hence "y  snd (F (?x, y))" using F_y by simp
hence 2: "y  ?y" by (simp add: gfp_upperbound)
show "z  (?x, ?y)" using z 1 2 by simp
qed

end