Theory Porder

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theory Porder
imports Main

(*  Title:      HOLCF/Porder.thy
    Author:     Franz Regensburger and Brian Huffman
*)

header {* Partial orders *}

theory Porder
imports Main
begin

subsection {* Type class for partial orders *}

class sq_ord =
  fixes sq_le :: "'a => 'a => bool"

notation
  sq_le (infixl "<<" 55)

notation (xsymbols)
  sq_le (infixl "\<sqsubseteq>" 55)

class po = sq_ord +
  assumes refl_less [iff]: "x \<sqsubseteq> x"
  assumes trans_less: "[|x \<sqsubseteq> y; y \<sqsubseteq> z|] ==> x \<sqsubseteq> z"
  assumes antisym_less: "[|x \<sqsubseteq> y; y \<sqsubseteq> x|] ==> x = y"

text {* minimal fixes least element *}

lemma minimal2UU[OF allI] : "∀x::'a::po. uu \<sqsubseteq> x ==> uu = (THE u. ∀y. u \<sqsubseteq> y)"
by (blast intro: theI2 antisym_less)

text {* the reverse law of anti-symmetry of @{term "op <<"} *}

lemma antisym_less_inverse: "(x::'a::po) = y ==> x \<sqsubseteq> y ∧ y \<sqsubseteq> x"
by simp

lemma box_less: "[|(a::'a::po) \<sqsubseteq> b; c \<sqsubseteq> a; b \<sqsubseteq> d|] ==> c \<sqsubseteq> d"
by (rule trans_less [OF trans_less])

lemma po_eq_conv: "((x::'a::po) = y) = (x \<sqsubseteq> y ∧ y \<sqsubseteq> x)"
by (fast elim!: antisym_less_inverse intro!: antisym_less)

lemma rev_trans_less: "[|(y::'a::po) \<sqsubseteq> z; x \<sqsubseteq> y|] ==> x \<sqsubseteq> z"
by (rule trans_less)

lemma sq_ord_less_eq_trans: "[|a \<sqsubseteq> b; b = c|] ==> a \<sqsubseteq> c"
by (rule subst)

lemma sq_ord_eq_less_trans: "[|a = b; b \<sqsubseteq> c|] ==> a \<sqsubseteq> c"
by (rule ssubst)

lemmas HOLCF_trans_rules [trans] =
  trans_less
  antisym_less
  sq_ord_less_eq_trans
  sq_ord_eq_less_trans

subsection {* Upper bounds *}

definition
  is_ub :: "['a set, 'a::po] => bool"  (infixl "<|" 55)  where
  "(S <| x) = (∀y. y ∈ S --> y \<sqsubseteq> x)"

lemma is_ubI: "(!!x. x ∈ S ==> x \<sqsubseteq> u) ==> S <| u"
by (simp add: is_ub_def)

lemma is_ubD: "[|S <| u; x ∈ S|] ==> x \<sqsubseteq> u"
by (simp add: is_ub_def)

lemma ub_imageI: "(!!x. x ∈ S ==> f x \<sqsubseteq> u) ==> (λx. f x) ` S <| u"
unfolding is_ub_def by fast

lemma ub_imageD: "[|f ` S <| u; x ∈ S|] ==> f x \<sqsubseteq> u"
unfolding is_ub_def by fast

lemma ub_rangeI: "(!!i. S i \<sqsubseteq> x) ==> range S <| x"
unfolding is_ub_def by fast

lemma ub_rangeD: "range S <| x ==> S i \<sqsubseteq> x"
unfolding is_ub_def by fast

lemma is_ub_empty [simp]: "{} <| u"
unfolding is_ub_def by fast

lemma is_ub_insert [simp]: "(insert x A) <| y = (x \<sqsubseteq> y ∧ A <| y)"
unfolding is_ub_def by fast

lemma is_ub_upward: "[|S <| x; x \<sqsubseteq> y|] ==> S <| y"
unfolding is_ub_def by (fast intro: trans_less)

subsection {* Least upper bounds *}

definition
  is_lub :: "['a set, 'a::po] => bool"  (infixl "<<|" 55)  where
  "(S <<| x) = (S <| x ∧ (∀u. S <| u --> x \<sqsubseteq> u))"

definition
  lub :: "'a set => 'a::po" where
  "lub S = (THE x. S <<| x)"

syntax
  "_BLub" :: "[pttrn, 'a set, 'b] => 'b" ("(3LUB _:_./ _)" [0,0, 10] 10)

syntax (xsymbols)
  "_BLub" :: "[pttrn, 'a set, 'b] => 'b" ("(3\<Squnion>_∈_./ _)" [0,0, 10] 10)

translations
  "LUB x:A. t" == "CONST lub ((%x. t) ` A)"

abbreviation
  Lub  (binder "LUB " 10) where
  "LUB n. t n == lub (range t)"

notation (xsymbols)
  Lub  (binder "\<Squnion> " 10)

text {* access to some definition as inference rule *}

lemma is_lubD1: "S <<| x ==> S <| x"
unfolding is_lub_def by fast

lemma is_lub_lub: "[|S <<| x; S <| u|] ==> x \<sqsubseteq> u"
unfolding is_lub_def by fast

lemma is_lubI: "[|S <| x; !!u. S <| u ==> x \<sqsubseteq> u|] ==> S <<| x"
unfolding is_lub_def by fast

text {* lubs are unique *}

lemma unique_lub: "[|S <<| x; S <<| y|] ==> x = y"
apply (unfold is_lub_def is_ub_def)
apply (blast intro: antisym_less)
done

text {* technical lemmas about @{term lub} and @{term is_lub} *}

lemma lubI: "M <<| x ==> M <<| lub M"
apply (unfold lub_def)
apply (rule theI)
apply assumption
apply (erule (1) unique_lub)
done

lemma thelubI: "M <<| l ==> lub M = l"
by (rule unique_lub [OF lubI])

lemma is_lub_singleton: "{x} <<| x"
by (simp add: is_lub_def)

lemma lub_singleton [simp]: "lub {x} = x"
by (rule thelubI [OF is_lub_singleton])

lemma is_lub_bin: "x \<sqsubseteq> y ==> {x, y} <<| y"
by (simp add: is_lub_def)

lemma lub_bin: "x \<sqsubseteq> y ==> lub {x, y} = y"
by (rule is_lub_bin [THEN thelubI])

lemma is_lub_maximal: "[|S <| x; x ∈ S|] ==> S <<| x"
by (erule is_lubI, erule (1) is_ubD)

lemma lub_maximal: "[|S <| x; x ∈ S|] ==> lub S = x"
by (rule is_lub_maximal [THEN thelubI])

subsection {* Countable chains *}

definition
  -- {* Here we use countable chains and I prefer to code them as functions! *}
  chain :: "(nat => 'a::po) => bool" where
  "chain Y = (∀i. Y i \<sqsubseteq> Y (Suc i))"

lemma chainI: "(!!i. Y i \<sqsubseteq> Y (Suc i)) ==> chain Y"
unfolding chain_def by fast

lemma chainE: "chain Y ==> Y i \<sqsubseteq> Y (Suc i)"
unfolding chain_def by fast

text {* chains are monotone functions *}

lemma chain_mono_less: "[|chain Y; i < j|] ==> Y i \<sqsubseteq> Y j"
by (erule less_Suc_induct, erule chainE, erule trans_less)

lemma chain_mono: "[|chain Y; i ≤ j|] ==> Y i \<sqsubseteq> Y j"
by (cases "i = j", simp, simp add: chain_mono_less)

lemma chain_shift: "chain Y ==> chain (λi. Y (i + j))"
by (rule chainI, simp, erule chainE)

text {* technical lemmas about (least) upper bounds of chains *}

lemma is_ub_lub: "range S <<| x ==> S i \<sqsubseteq> x"
by (rule is_lubD1 [THEN ub_rangeD])

lemma is_ub_range_shift:
  "chain S ==> range (λi. S (i + j)) <| x = range S <| x"
apply (rule iffI)
apply (rule ub_rangeI)
apply (rule_tac y="S (i + j)" in trans_less)
apply (erule chain_mono)
apply (rule le_add1)
apply (erule ub_rangeD)
apply (rule ub_rangeI)
apply (erule ub_rangeD)
done

lemma is_lub_range_shift:
  "chain S ==> range (λi. S (i + j)) <<| x = range S <<| x"
by (simp add: is_lub_def is_ub_range_shift)

text {* the lub of a constant chain is the constant *}

lemma chain_const [simp]: "chain (λi. c)"
by (simp add: chainI)

lemma lub_const: "range (λx. c) <<| c"
by (blast dest: ub_rangeD intro: is_lubI ub_rangeI)

lemma thelub_const [simp]: "(\<Squnion>i. c) = c"
by (rule lub_const [THEN thelubI])

subsection {* Finite chains *}

definition
  -- {* finite chains, needed for monotony of continuous functions *}
  max_in_chain :: "[nat, nat => 'a::po] => bool" where
  "max_in_chain i C = (∀j. i ≤ j --> C i = C j)"

definition
  finite_chain :: "(nat => 'a::po) => bool" where
  "finite_chain C = (chain C ∧ (∃i. max_in_chain i C))"

text {* results about finite chains *}

lemma max_in_chainI: "(!!j. i ≤ j ==> Y i = Y j) ==> max_in_chain i Y"
unfolding max_in_chain_def by fast

lemma max_in_chainD: "[|max_in_chain i Y; i ≤ j|] ==> Y i = Y j"
unfolding max_in_chain_def by fast

lemma finite_chainI:
  "[|chain C; max_in_chain i C|] ==> finite_chain C"
unfolding finite_chain_def by fast

lemma finite_chainE:
  "[|finite_chain C; !!i. [|chain C; max_in_chain i C|] ==> R|] ==> R"
unfolding finite_chain_def by fast

lemma lub_finch1: "[|chain C; max_in_chain i C|] ==> range C <<| C i"
apply (rule is_lubI)
apply (rule ub_rangeI, rename_tac j)
apply (rule_tac x=i and y=j in linorder_le_cases)
apply (drule (1) max_in_chainD, simp)
apply (erule (1) chain_mono)
apply (erule ub_rangeD)
done

lemma lub_finch2:
  "finite_chain C ==> range C <<| C (LEAST i. max_in_chain i C)"
apply (erule finite_chainE)
apply (erule LeastI2 [where Q="λi. range C <<| C i"])
apply (erule (1) lub_finch1)
done

lemma finch_imp_finite_range: "finite_chain Y ==> finite (range Y)"
 apply (erule finite_chainE)
 apply (rule_tac B="Y ` {..i}" in finite_subset)
  apply (rule subsetI)
  apply (erule rangeE, rename_tac j)
  apply (rule_tac x=i and y=j in linorder_le_cases)
   apply (subgoal_tac "Y j = Y i", simp)
   apply (simp add: max_in_chain_def)
  apply simp
 apply simp
done

lemma finite_range_has_max:
  fixes f :: "nat => 'a" and r :: "'a => 'a => bool"
  assumes mono: "!!i j. i ≤ j ==> r (f i) (f j)"
  assumes finite_range: "finite (range f)"
  shows "∃k. ∀i. r (f i) (f k)"
proof (intro exI allI)
  fix i :: nat
  let ?j = "LEAST k. f k = f i"
  let ?k = "Max ((λx. LEAST k. f k = x) ` range f)"
  have "?j ≤ ?k"
  proof (rule Max_ge)
    show "finite ((λx. LEAST k. f k = x) ` range f)"
      using finite_range by (rule finite_imageI)
    show "?j ∈ (λx. LEAST k. f k = x) ` range f"
      by (intro imageI rangeI)
  qed
  hence "r (f ?j) (f ?k)"
    by (rule mono)
  also have "f ?j = f i"
    by (rule LeastI, rule refl)
  finally show "r (f i) (f ?k)" .
qed

lemma finite_range_imp_finch:
  "[|chain Y; finite (range Y)|] ==> finite_chain Y"
 apply (subgoal_tac "∃k. ∀i. Y i \<sqsubseteq> Y k")
  apply (erule exE)
  apply (rule finite_chainI, assumption)
  apply (rule max_in_chainI)
  apply (rule antisym_less)
   apply (erule (1) chain_mono)
  apply (erule spec)
 apply (rule finite_range_has_max)
  apply (erule (1) chain_mono)
 apply assumption
done

lemma bin_chain: "x \<sqsubseteq> y ==> chain (λi. if i=0 then x else y)"
by (rule chainI, simp)

lemma bin_chainmax:
  "x \<sqsubseteq> y ==> max_in_chain (Suc 0) (λi. if i=0 then x else y)"
unfolding max_in_chain_def by simp

lemma lub_bin_chain:
  "x \<sqsubseteq> y ==> range (λi::nat. if i=0 then x else y) <<| y"
apply (frule bin_chain)
apply (drule bin_chainmax)
apply (drule (1) lub_finch1)
apply simp
done

text {* the maximal element in a chain is its lub *}

lemma lub_chain_maxelem: "[|Y i = c; ∀i. Y i \<sqsubseteq> c|] ==> lub (range Y) = c"
by (blast dest: ub_rangeD intro: thelubI is_lubI ub_rangeI)

subsection {* Directed sets *}

definition
  directed :: "'a::po set => bool" where
  "directed S = ((∃x. x ∈ S) ∧ (∀x∈S. ∀y∈S. ∃z∈S. x \<sqsubseteq> z ∧ y \<sqsubseteq> z))"

lemma directedI:
  assumes 1: "∃z. z ∈ S"
  assumes 2: "!!x y. [|x ∈ S; y ∈ S|] ==> ∃z∈S. x \<sqsubseteq> z ∧ y \<sqsubseteq> z"
  shows "directed S"
unfolding directed_def using prems by fast

lemma directedD1: "directed S ==> ∃z. z ∈ S"
unfolding directed_def by fast

lemma directedD2: "[|directed S; x ∈ S; y ∈ S|] ==> ∃z∈S. x \<sqsubseteq> z ∧ y \<sqsubseteq> z"
unfolding directed_def by fast

lemma directedE1:
  assumes S: "directed S"
  obtains z where "z ∈ S"
by (insert directedD1 [OF S], fast)

lemma directedE2:
  assumes S: "directed S"
  assumes x: "x ∈ S" and y: "y ∈ S"
  obtains z where "z ∈ S" "x \<sqsubseteq> z" "y \<sqsubseteq> z"
by (insert directedD2 [OF S x y], fast)

lemma directed_finiteI:
  assumes U: "!!U. [|finite U; U ⊆ S|] ==> ∃z∈S. U <| z"
  shows "directed S"
proof (rule directedI)
  have "finite {}" and "{} ⊆ S" by simp_all
  hence "∃z∈S. {} <| z" by (rule U)
  thus "∃z. z ∈ S" by simp
next
  fix x y
  assume "x ∈ S" and "y ∈ S"
  hence "finite {x, y}" and "{x, y} ⊆ S" by simp_all
  hence "∃z∈S. {x, y} <| z" by (rule U)
  thus "∃z∈S. x \<sqsubseteq> z ∧ y \<sqsubseteq> z" by simp
qed

lemma directed_finiteD:
  assumes S: "directed S"
  shows "[|finite U; U ⊆ S|] ==> ∃z∈S. U <| z"
proof (induct U set: finite)
  case empty
  from S have "∃z. z ∈ S" by (rule directedD1)
  thus "∃z∈S. {} <| z" by simp
next
  case (insert x F)
  from `insert x F ⊆ S`
  have xS: "x ∈ S" and FS: "F ⊆ S" by simp_all
  from FS have "∃y∈S. F <| y" by fact
  then obtain y where yS: "y ∈ S" and Fy: "F <| y" ..
  obtain z where zS: "z ∈ S" and xz: "x \<sqsubseteq> z" and yz: "y \<sqsubseteq> z"
    using S xS yS by (rule directedE2)
  from Fy yz have "F <| z" by (rule is_ub_upward)
  with xz have "insert x F <| z" by simp
  with zS show "∃z∈S. insert x F <| z" ..
qed

lemma not_directed_empty [simp]: "¬ directed {}"
by (rule notI, drule directedD1, simp)

lemma directed_singleton: "directed {x}"
by (rule directedI, auto)

lemma directed_bin: "x \<sqsubseteq> y ==> directed {x, y}"
by (rule directedI, auto)

lemma directed_chain: "chain S ==> directed (range S)"
apply (rule directedI)
apply (rule_tac x="S 0" in exI, simp)
apply (clarify, rename_tac m n)
apply (rule_tac x="S (max m n)" in bexI)
apply (simp add: chain_mono)
apply simp
done

end