# Theory HOL-Library.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

lemma atLeastAtMost_prod_eq: "{a..b} = {fst a..fst b} × {snd a..snd b}"
by (auto simp: less_eq_prod_def)

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]: "fst top = top"
unfolding top_prod_def by simp

lemma snd_top [simp]: "snd top = top"
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]: "fst bot = bot"
unfolding bot_prod_def by simp

lemma snd_bot [simp]: "snd bot = bot"
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 x∈A. fst x, INF x∈A. snd x)"

instance ..

end

instantiation prod :: (Sup, Sup) Sup
begin

definition "Sup A = (SUP x∈A. fst x, SUP x∈A. 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 x∈A. fst x)"
by (simp add: Inf_prod_def)

lemma fst_INF: "fst (INF x∈A. f x) = (INF x∈A. fst (f x))"
by (simp add: fst_Inf image_image)

lemma fst_Sup: "fst (Sup A) = (SUP x∈A. fst x)"
by (simp add: Sup_prod_def)

lemma fst_SUP: "fst (SUP x∈A. f x) = (SUP x∈A. fst (f x))"
by (simp add: fst_Sup image_image)

lemma snd_Inf: "snd (Inf A) = (INF x∈A. snd x)"
by (simp add: Inf_prod_def)

lemma snd_INF: "snd (INF x∈A. f x) = (INF x∈A. snd (f x))"
by (simp add: snd_Inf image_image)

lemma snd_Sup: "snd (Sup A) = (SUP x∈A. snd x)"
by (simp add: Sup_prod_def)

lemma snd_SUP: "snd (SUP x∈A. f x) = (SUP x∈A. snd (f x))"
by (simp add: snd_Sup image_image)

lemma INF_Pair: "(INF x∈A. (f x, g x)) = (INF x∈A. f x, INF x∈A. g x)"
by (simp add: Inf_prod_def image_image)

lemma SUP_Pair: "(SUP x∈A. (f x, g x)) = (SUP x∈A. f x, SUP x∈A. g x)"
by (simp add: Sup_prod_def image_image)

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))"
by (simp add: Inf_prod_def image_image)

lemma SUP_prod_alt_def: ✐‹contributor ‹Alessandro Coglio››
"Sup (f ` A) = (Sup ((fst ∘ f) ` A), Sup((snd ∘ f) ` A))"
by (simp add: Sup_prod_def image_image)

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. ∀Y∈A. 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
```