# Theory Semilat

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theory Semilat
imports While_Combinator
(*  Title:      HOL/MicroJava/DFA/Semilat.thy
Author: Tobias Nipkow
*)

\chapter{Bytecode Verifier}\label{cha:bv}
*}

theory Semilat
imports Main "~~/src/HOL/Library/While_Combinator"
begin

type_synonym 'a ord = "'a => 'a => bool"
type_synonym 'a binop = "'a => 'a => 'a"
type_synonym 'a sl = "'a set × 'a ord × 'a binop"

consts
"lesub" :: "'a => 'a ord => 'a => bool"
"lesssub" :: "'a => 'a ord => 'a => bool"
"plussub" :: "'a => ('a => 'b => 'c) => 'b => 'c"
(*<*)
notation
"lesub" ("(_ /<='__ _)" [50, 1000, 51] 50) and
"lesssub" ("(_ /<'__ _)" [50, 1000, 51] 50) and
"plussub" ("(_ /+'__ _)" [65, 1000, 66] 65)
(*>*)
notation (xsymbols)
"lesub" ("(_ /\<sqsubseteq>_ _)" [50, 0, 51] 50) and
"lesssub" ("(_ /\<sqsubset>_ _)" [50, 0, 51] 50) and
"plussub" ("(_ /\<squnion>_ _)" [65, 0, 66] 65)
(*<*)
(* allow \<sub> instead of \<bsub>..\<esub> *)

abbreviation (input)
lesub1 :: "'a => 'a ord => 'a => bool" ("(_ /\<sqsubseteq>_ _)" [50, 1000, 51] 50)
where "x \<sqsubseteq>r y == x \<sqsubseteq>r y"

abbreviation (input)
lesssub1 :: "'a => 'a ord => 'a => bool" ("(_ /\<sqsubset>_ _)" [50, 1000, 51] 50)
where "x \<sqsubset>r y == x \<sqsubset>r y"

abbreviation (input)
plussub1 :: "'a => ('a => 'b => 'c) => 'b => 'c" ("(_ /\<squnion>_ _)" [65, 1000, 66] 65)
where "x \<squnion>f y == x \<squnion>f y"
(*>*)

defs
lesub_def: "x \<sqsubseteq>r y ≡ r x y"
lesssub_def: "x \<sqsubset>r y ≡ x \<sqsubseteq>r y ∧ x ≠ y"
plussub_def: "x \<squnion>f y ≡ f x y"

definition ord :: "('a × 'a) set => 'a ord" where
"ord r ≡ λx y. (x,y) ∈ r"

definition order :: "'a ord => bool" where
"order r ≡ (∀x. x \<sqsubseteq>r x) ∧ (∀x y. x \<sqsubseteq>r y ∧ y \<sqsubseteq>r x --> x=y) ∧ (∀x y z. x \<sqsubseteq>r y ∧ y \<sqsubseteq>r z --> x \<sqsubseteq>r z)"

definition top :: "'a ord => 'a => bool" where
"top r T ≡ ∀x. x \<sqsubseteq>r T"

definition acc :: "'a ord => bool" where
"acc r ≡ wf {(y,x). x \<sqsubset>r y}"

definition closed :: "'a set => 'a binop => bool" where
"closed A f ≡ ∀x∈A. ∀y∈A. x \<squnion>f y ∈ A"

definition semilat :: "'a sl => bool" where
"semilat ≡ λ(A,r,f). order r ∧ closed A f ∧
(∀x∈A. ∀y∈A. x \<sqsubseteq>r x \<squnion>f y) ∧
(∀x∈A. ∀y∈A. y \<sqsubseteq>r x \<squnion>f y) ∧
(∀x∈A. ∀y∈A. ∀z∈A. x \<sqsubseteq>r z ∧ y \<sqsubseteq>r z --> x \<squnion>f y \<sqsubseteq>r z)"

definition is_ub :: "('a × 'a) set => 'a => 'a => 'a => bool" where
"is_ub r x y u ≡ (x,u)∈r ∧ (y,u)∈r"

definition is_lub :: "('a × 'a) set => 'a => 'a => 'a => bool" where
"is_lub r x y u ≡ is_ub r x y u ∧ (∀z. is_ub r x y z --> (u,z)∈r)"

definition some_lub :: "('a × 'a) set => 'a => 'a => 'a" where
"some_lub r x y ≡ SOME z. is_lub r x y z"

locale Semilat =
fixes A :: "'a set"
fixes r :: "'a ord"
fixes f :: "'a binop"
assumes semilat: "semilat (A, r, f)"

lemma order_refl [simp, intro]: "order r ==> x \<sqsubseteq>r x"
(*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)

lemma order_antisym: "[| order r; x \<sqsubseteq>r y; y \<sqsubseteq>r x |] ==> x = y"
(*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)

lemma order_trans: "[| order r; x \<sqsubseteq>r y; y \<sqsubseteq>r z |] ==> x \<sqsubseteq>r z"
(*<*) by (unfold order_def) blast (*>*)

lemma order_less_irrefl [intro, simp]: "order r ==> ¬ x \<sqsubset>r x"
(*<*) by (unfold order_def lesssub_def) blast (*>*)

lemma order_less_trans: "[| order r; x \<sqsubset>r y; y \<sqsubset>r z |] ==> x \<sqsubset>r z"
(*<*) by (unfold order_def lesssub_def) blast (*>*)

lemma topD [simp, intro]: "top r T ==> x \<sqsubseteq>r T"
(*<*) by (simp add: top_def) (*>*)

lemma top_le_conv [simp]: "[| order r; top r T |] ==> (T \<sqsubseteq>r x) = (x = T)"
(*<*) by (blast intro: order_antisym) (*>*)

lemma semilat_Def:
"semilat(A,r,f) ≡ order r ∧ closed A f ∧
(∀x∈A. ∀y∈A. x \<sqsubseteq>r x \<squnion>f y) ∧
(∀x∈A. ∀y∈A. y \<sqsubseteq>r x \<squnion>f y) ∧
(∀x∈A. ∀y∈A. ∀z∈A. x \<sqsubseteq>r z ∧ y \<sqsubseteq>r z --> x \<squnion>f y \<sqsubseteq>r z)"

(*<*) by (unfold semilat_def) clarsimp (*>*)

lemma (in Semilat) orderI [simp, intro]: "order r"
(*<*) using semilat by (simp add: semilat_Def) (*>*)

lemma (in Semilat) closedI [simp, intro]: "closed A f"
(*<*) using semilat by (simp add: semilat_Def) (*>*)

lemma closedD: "[| closed A f; x∈A; y∈A |] ==> x \<squnion>f y ∈ A"
(*<*) by (unfold closed_def) blast (*>*)

lemma closed_UNIV [simp]: "closed UNIV f"
(*<*) by (simp add: closed_def) (*>*)

lemma (in Semilat) closed_f [simp, intro]: "[|x ∈ A; y ∈ A|] ==> x \<squnion>f y ∈ A"
(*<*) by (simp add: closedD [OF closedI]) (*>*)

lemma (in Semilat) refl_r [intro, simp]: "x \<sqsubseteq>r x" by simp

lemma (in Semilat) antisym_r [intro?]: "[| x \<sqsubseteq>r y; y \<sqsubseteq>r x |] ==> x = y"
(*<*) by (rule order_antisym) auto (*>*)

lemma (in Semilat) trans_r [trans, intro?]: "[|x \<sqsubseteq>r y; y \<sqsubseteq>r z|] ==> x \<sqsubseteq>r z"
(*<*) by (auto intro: order_trans) (*>*)

lemma (in Semilat) ub1 [simp, intro?]: "[| x ∈ A; y ∈ A |] ==> x \<sqsubseteq>r x \<squnion>f y"
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)

lemma (in Semilat) ub2 [simp, intro?]: "[| x ∈ A; y ∈ A |] ==> y \<sqsubseteq>r x \<squnion>f y"
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)

lemma (in Semilat) lub [simp, intro?]:
"[| x \<sqsubseteq>r z; y \<sqsubseteq>r z; x ∈ A; y ∈ A; z ∈ A |] ==> x \<squnion>f y \<sqsubseteq>r z";
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)

lemma (in Semilat) plus_le_conv [simp]:
"[| x ∈ A; y ∈ A; z ∈ A |] ==> (x \<squnion>f y \<sqsubseteq>r z) = (x \<sqsubseteq>r z ∧ y \<sqsubseteq>r z)"
(*<*) by (blast intro: ub1 ub2 lub order_trans) (*>*)

lemma (in Semilat) le_iff_plus_unchanged: "[| x ∈ A; y ∈ A |] ==> (x \<sqsubseteq>r y) = (x \<squnion>f y = y)"
(*<*)
apply (rule iffI)
apply (blast intro: antisym_r lub ub2)
apply (erule subst)
apply simp
done
(*>*)

lemma (in Semilat) le_iff_plus_unchanged2: "[| x ∈ A; y ∈ A |] ==> (x \<sqsubseteq>r y) = (y \<squnion>f x = y)"
(*<*)
apply (rule iffI)
apply (blast intro: order_antisym lub ub1)
apply (erule subst)
apply simp
done
(*>*)

lemma (in Semilat) plus_assoc [simp]:
assumes a: "a ∈ A" and b: "b ∈ A" and c: "c ∈ A"
shows "a \<squnion>f (b \<squnion>f c) = a \<squnion>f b \<squnion>f c"
(*<*)
proof -
from a b have ab: "a \<squnion>f b ∈ A" ..
from this c have abc: "(a \<squnion>f b) \<squnion>f c ∈ A" ..
from b c have bc: "b \<squnion>f c ∈ A" ..
from a this have abc': "a \<squnion>f (b \<squnion>f c) ∈ A" ..

show ?thesis
proof
show "a \<squnion>f (b \<squnion>f c) \<sqsubseteq>r (a \<squnion>f b) \<squnion>f c"
proof -
from a b have "a \<sqsubseteq>r a \<squnion>f b" ..
also from ab c have "… \<sqsubseteq>r … \<squnion>f c" ..
finally have "a<": "a \<sqsubseteq>r (a \<squnion>f b) \<squnion>f c" .
from a b have "b \<sqsubseteq>r a \<squnion>f b" ..
also from ab c have "… \<sqsubseteq>r … \<squnion>f c" ..
finally have "b<": "b \<sqsubseteq>r (a \<squnion>f b) \<squnion>f c" .
from ab c have "c<": "c \<sqsubseteq>r (a \<squnion>f b) \<squnion>f c" ..
from "b<" "c<" b c abc have "b \<squnion>f c \<sqsubseteq>r (a \<squnion>f b) \<squnion>f c" ..
from "a<" this a bc abc show ?thesis ..
qed
show "(a \<squnion>f b) \<squnion>f c \<sqsubseteq>r a \<squnion>f (b \<squnion>f c)"
proof -
from b c have "b \<sqsubseteq>r b \<squnion>f c" ..
also from a bc have "… \<sqsubseteq>r a \<squnion>f …" ..
finally have "b<": "b \<sqsubseteq>r a \<squnion>f (b \<squnion>f c)" .
from b c have "c \<sqsubseteq>r b \<squnion>f c" ..
also from a bc have "… \<sqsubseteq>r a \<squnion>f …" ..
finally have "c<": "c \<sqsubseteq>r a \<squnion>f (b \<squnion>f c)" .
from a bc have "a<": "a \<sqsubseteq>r a \<squnion>f (b \<squnion>f c)" ..
from "a<" "b<" a b abc' have "a \<squnion>f b \<sqsubseteq>r a \<squnion>f (b \<squnion>f c)" ..
from this "c<" ab c abc' show ?thesis ..
qed
qed
qed
(*>*)

lemma (in Semilat) plus_com_lemma:
"[|a ∈ A; b ∈ A|] ==> a \<squnion>f b \<sqsubseteq>r b \<squnion>f a"
(*<*)
proof -
assume a: "a ∈ A" and b: "b ∈ A"
from b a have "a \<sqsubseteq>r b \<squnion>f a" ..
moreover from b a have "b \<sqsubseteq>r b \<squnion>f a" ..
moreover note a b
moreover from b a have "b \<squnion>f a ∈ A" ..
ultimately show ?thesis ..
qed
(*>*)

lemma (in Semilat) plus_commutative:
"[|a ∈ A; b ∈ A|] ==> a \<squnion>f b = b \<squnion>f a"
(*<*) by(blast intro: order_antisym plus_com_lemma) (*>*)

lemma is_lubD:
"is_lub r x y u ==> is_ub r x y u ∧ (∀z. is_ub r x y z --> (u,z) ∈ r)"
(*<*) by (simp add: is_lub_def) (*>*)

lemma is_ubI:
"[| (x,u) ∈ r; (y,u) ∈ r |] ==> is_ub r x y u"
(*<*) by (simp add: is_ub_def) (*>*)

lemma is_ubD:
"is_ub r x y u ==> (x,u) ∈ r ∧ (y,u) ∈ r"
(*<*) by (simp add: is_ub_def) (*>*)

lemma is_lub_bigger1 [iff]:
"is_lub (r^* ) x y y = ((x,y)∈r^* )"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply blast
done
(*>*)

lemma is_lub_bigger2 [iff]:
"is_lub (r^* ) x y x = ((y,x)∈r^* )"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply blast
done
(*>*)

lemma extend_lub:
"[| single_valued r; is_lub (r^* ) x y u; (x',x) ∈ r |]
==> EX v. is_lub (r^* ) x' y v"

(*<*)
apply (unfold is_lub_def is_ub_def)
apply (case_tac "(y,x) ∈ r^*")
apply (case_tac "(y,x') ∈ r^*")
apply blast
apply (blast elim: converse_rtranclE dest: single_valuedD)
apply (rule exI)
apply (rule conjI)
apply (blast intro: converse_rtrancl_into_rtrancl dest: single_valuedD)
apply (blast intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
elim: converse_rtranclE dest: single_valuedD)
done
(*>*)

lemma single_valued_has_lubs [rule_format]:
"[| single_valued r; (x,u) ∈ r^* |] ==> (∀y. (y,u) ∈ r^* -->
(EX z. is_lub (r^* ) x y z))"

(*<*)
apply (erule converse_rtrancl_induct)
apply clarify
apply (erule converse_rtrancl_induct)
apply blast
apply (blast intro: converse_rtrancl_into_rtrancl)
apply (blast intro: extend_lub)
done
(*>*)

lemma some_lub_conv:
"[| acyclic r; is_lub (r^* ) x y u |] ==> some_lub (r^* ) x y = u"
(*<*)
apply (unfold some_lub_def is_lub_def)
apply (rule someI2)
apply assumption
apply (blast intro: antisymD dest!: acyclic_impl_antisym_rtrancl)
done
(*>*)

lemma is_lub_some_lub:
"[| single_valued r; acyclic r; (x,u)∈r^*; (y,u)∈r^* |]
==> is_lub (r^* ) x y (some_lub (r^* ) x y)"
;
(*<*) by (fastforce dest: single_valued_has_lubs simp add: some_lub_conv) (*>*)

subsection{*An executable lub-finder*}

definition exec_lub :: "('a * 'a) set => ('a => 'a) => 'a binop" where
"exec_lub r f x y ≡ while (λz. (x,z) ∉ r*) f y"

lemma exec_lub_refl: "exec_lub r f T T = T"

lemma acyclic_single_valued_finite:
"[|acyclic r; single_valued r; (x,y) ∈ r*|]
==> finite (r ∩ {a. (x, a) ∈ r*} × {b. (b, y) ∈ r*})"

(*<*)
apply(erule converse_rtrancl_induct)
apply(rule_tac B = "{}" in finite_subset)
apply(simp only:acyclic_def)
apply(blast intro:rtrancl_into_trancl2 rtrancl_trancl_trancl)
apply simp
apply(rename_tac x x')
apply(subgoal_tac "r ∩ {a. (x,a) ∈ r*} × {b. (b,y) ∈ r*} =
insert (x,x') (r ∩ {a. (x', a) ∈ r*} × {b. (b, y) ∈ r*})"
)
apply simp
apply(blast intro:converse_rtrancl_into_rtrancl
elim:converse_rtranclE dest:single_valuedD)
done
(*>*)

lemma exec_lub_conv:
"[| acyclic r; ∀x y. (x,y) ∈ r --> f x = y; is_lub (r*) x y u |] ==>
exec_lub r f x y = u"
;
(*<*)
apply(unfold exec_lub_def)
apply(rule_tac P = "λz. (y,z) ∈ r* ∧ (z,u) ∈ r*" and
r = "(r ∩ {(a,b). (y,a) ∈ r* ∧ (b,u) ∈ r*})^-1" in while_rule)
apply(blast dest: is_lubD is_ubD)
apply(erule conjE)
apply(erule_tac z = u in converse_rtranclE)
apply(blast dest: is_lubD is_ubD)
apply(blast dest:rtrancl_into_rtrancl)
apply(rename_tac s)
apply(subgoal_tac "is_ub (r*) x y s")
apply(subgoal_tac "(u, s) ∈ r*")
prefer 2; apply(blast dest:is_lubD)
apply(erule converse_rtranclE)
apply blast
apply(simp only:acyclic_def)
apply(blast intro:rtrancl_into_trancl2 rtrancl_trancl_trancl)
apply(rule finite_acyclic_wf)
apply simp
apply(erule acyclic_single_valued_finite)
apply(blast intro:single_valuedI)
apply simp
apply(erule acyclic_subset)
apply blast
apply simp
apply(erule conjE)
apply(erule_tac z = u in converse_rtranclE)
apply(blast dest: is_lubD is_ubD)
apply(blast dest:rtrancl_into_rtrancl)
done
(*>*)

lemma is_lub_exec_lub:
"[| single_valued r; acyclic r; (x,u):r^*; (y,u):r^*; ∀x y. (x,y) ∈ r --> f x = y |]
==> is_lub (r^* ) x y (exec_lub r f x y)"

(*<*) by (fastforce dest: single_valued_has_lubs simp add: exec_lub_conv) (*>*)

end