# Theory Follows

theory Follows
imports SubstAx Increasing
```(*  Title:      ZF/UNITY/Follows.thy
Author:     Sidi O Ehmety, Cambridge University Computer Laboratory

Theory ported from HOL.
*)

section‹The "Follows" relation of Charpentier and Sivilotte›

theory Follows imports SubstAx Increasing begin

definition
Follows :: "[i, i, i=>i, i=>i] => i"  where
"Follows(A, r, f, g) ==
Increasing(A, r, g) Int
Increasing(A, r,f) Int
Always({s ∈ state. <f(s), g(s)>:r}) Int
(⋂k ∈ A. {s ∈ state. <k, g(s)>:r} ⟼w {s ∈ state. <k,f(s)>:r})"

abbreviation
Incr :: "[i=>i]=>i" where
"Incr(f) == Increasing(list(nat), prefix(nat), f)"

abbreviation
n_Incr :: "[i=>i]=>i" where
"n_Incr(f) == Increasing(nat, Le, f)"

abbreviation
s_Incr :: "[i=>i]=>i" where
"s_Incr(f) == Increasing(Pow(nat), SetLe(nat), f)"

abbreviation
m_Incr :: "[i=>i]=>i" where
"m_Incr(f) == Increasing(Mult(nat), MultLe(nat, Le), f)"

abbreviation
n_Fols :: "[i=>i, i=>i]=>i"   (infixl "n'_Fols" 65)  where
"f n_Fols g == Follows(nat, Le, f, g)"

abbreviation
Follows' :: "[i=>i, i=>i, i, i] => i"
("(_ /Fols _ /Wrt (_ /'/ _))" [60, 0, 0, 60] 60)  where
"f Fols g Wrt r/A == Follows(A,r,f,g)"

(*Does this hold for "invariant"?*)

lemma Follows_cong:
"[|A=A'; r=r'; !!x. x ∈ state ==> f(x)=f'(x); !!x. x ∈ state ==> g(x)=g'(x)|] ==> Follows(A, r, f, g) = Follows(A', r', f', g')"

lemma subset_Always_comp:
"[| mono1(A, r, B, s, h); ∀x ∈ state. f(x):A & g(x):A |] ==>
Always({x ∈ state. <f(x), g(x)> ∈ r})<=Always({x ∈ state. <(h comp f)(x), (h comp g)(x)> ∈ s})"
apply (unfold mono1_def metacomp_def)
done

lemma imp_Always_comp:
"[| F ∈ Always({x ∈ state. <f(x), g(x)> ∈ r});
mono1(A, r, B, s, h); ∀x ∈ state. f(x):A & g(x):A |] ==>
F ∈ Always({x ∈ state. <(h comp f)(x), (h comp g)(x)> ∈ s})"
by (blast intro: subset_Always_comp [THEN subsetD])

lemma imp_Always_comp2:
"[| F ∈ Always({x ∈ state. <f1(x), f(x)> ∈ r});
F ∈ Always({x ∈ state. <g1(x), g(x)> ∈ s});
mono2(A, r, B, s, C, t, h);
∀x ∈ state. f1(x):A & f(x):A & g1(x):B & g(x):B |]
==> F ∈ Always({x ∈ state. <h(f1(x), g1(x)), h(f(x), g(x))> ∈ t})"
apply (auto simp add: Always_eq_includes_reachable mono2_def)
apply (auto dest!: subsetD)
done

"[| mono1(A, r, B, s, h); refl(A,r); trans[B](s);
∀x ∈ state. f(x):A & g(x):A |] ==>
(⋂j ∈ A. {s ∈ state. <j, g(s)> ∈ r} ⟼w {s ∈ state. <j,f(s)> ∈ r}) <=
(⋂k ∈ B. {x ∈ state. <k, (h comp g)(x)> ∈ s} ⟼w {x ∈ state. <k, (h comp f)(x)> ∈ s})"

apply (unfold mono1_def metacomp_def, clarify)
apply auto
prefer 2 apply (blast dest: LeadsTo_type [THEN subsetD], auto)
apply (rotate_tac 5)
apply (drule_tac x = "g (sa) " in bspec)
apply (auto simp add: part_order_def refl_def)
apply (rule_tac b = "h (g (sa))" in trans_onD)
apply blast
apply auto
done

"[| F:(⋂j ∈ A. {s ∈ state. <j, g(s)> ∈ r} ⟼w {s ∈ state. <j,f(s)> ∈ r});
mono1(A, r, B, s, h); refl(A,r); trans[B](s);
∀x ∈ state. f(x):A & g(x):A |] ==>
F:(⋂k ∈ B. {x ∈ state. <k, (h comp g)(x)> ∈ s} ⟼w {x ∈ state. <k, (h comp f)(x)> ∈ s})"
apply (rule subset_LeadsTo_comp [THEN subsetD], auto)
done

"[| F ∈ Increasing(B, s, g);
∀j ∈ A. F: {s ∈ state. <j, f(s)> ∈ r} ⟼w {s ∈ state. <j,f1(s)> ∈ r};
mono2(A, r, B, s, C, t, h); refl(A, r); refl(B, s); trans[C](t);
∀x ∈ state. f1(x):A & f(x):A & g(x):B; k ∈ C |] ==>
F:{x ∈ state. <k, h(f(x), g(x))> ∈ t} ⟼w {x ∈ state. <k, h(f1(x), g(x))> ∈ t}"
apply (unfold mono2_def Increasing_def)
apply (drule_tac x = "g (sa) " and A = B in bspec)
apply auto
apply (drule_tac x = "f (sa) " and P = "%j. F ∈ X(j) ⟼w Y(j)" for X Y in bspec)
apply auto
apply (rule PSP_Stable [THEN LeadsTo_weaken], blast, blast)
apply auto
apply (force simp add: part_order_def refl_def)
apply (force simp add: part_order_def refl_def)
apply (drule_tac x = "f1 (x)" and x1 = "f (sa) " and P2 = "%x y. ∀u∈B. P (x,y,u)" for P in bspec [THEN bspec])
apply (drule_tac [3] x = "g (x) " and x1 = "g (sa) " and P2 = "%x y. P (x,y) ⟶ d (x,y) ∈ t" for P d in bspec [THEN bspec])
apply auto
apply (rule_tac b = "h (f (sa), g (sa))" and A = C in trans_onD)
done

"[| F ∈ Increasing(A, r, f);
∀j ∈ B. F: {x ∈ state. <j, g(x)> ∈ s} ⟼w {x ∈ state. <j,g1(x)> ∈ s};
mono2(A, r, B, s, C, t, h); refl(A,r); refl(B, s); trans[C](t);
∀x ∈ state. f(x):A & g1(x):B & g(x):B; k ∈ C |] ==>
F:{x ∈ state. <k, h(f(x), g(x))> ∈ t} ⟼w {x ∈ state. <k, h(f(x), g1(x))> ∈ t}"
apply (unfold mono2_def Increasing_def)
apply (drule_tac x = "f (sa) " and P = "%k. F ∈ Stable (X (k))" for X in bspec)
apply auto
apply (drule_tac x = "g (sa) " in bspec)
apply auto
apply (rule PSP_Stable [THEN LeadsTo_weaken], blast, blast)
apply auto
apply (force simp add: part_order_def refl_def)
apply (force simp add: part_order_def refl_def)
apply (drule_tac x = "f (x) " and x1 = "f (sa) " in bspec [THEN bspec])
apply (drule_tac [3] x = "g1 (x) " and x1 = "g (sa) " and P2 = "%x y. P (x,y) ⟶ d (x,y) ∈ t" for P d in bspec [THEN bspec])
apply auto
apply (rule_tac b = "h (f (sa), g (sa))" and A = C in trans_onD)
done

(**  This general result is used to prove Follows Un, munion, etc. **)
"[| F ∈ Increasing(A, r, f1) ∩  Increasing(B, s, g);
∀j ∈ A. F: {s ∈ state. <j, f(s)> ∈ r} ⟼w {s ∈ state. <j,f1(s)> ∈ r};
∀j ∈ B. F: {x ∈ state. <j, g(x)> ∈ s} ⟼w {x ∈ state. <j,g1(x)> ∈ s};
mono2(A, r, B, s, C, t, h); refl(A,r); refl(B, s); trans[C](t);
∀x ∈ state. f(x):A & g1(x):B & f1(x):A &g(x):B; k ∈ C |]
==> F:{x ∈ state. <k, h(f(x), g(x))> ∈ t} ⟼w {x ∈ state. <k, h(f1(x), g1(x))> ∈ t}"
apply (rule_tac B = "{x ∈ state. <k, h (f1 (x), g (x))> ∈ t}" in LeadsTo_Trans)
done

(* Follows type *)
lemma Follows_type: "Follows(A, r, f, g)<=program"
apply (unfold Follows_def)
apply (blast dest: Increasing_type [THEN subsetD])
done

lemma Follows_into_program [TC]: "F ∈ Follows(A, r, f, g) ==> F ∈ program"
by (blast dest: Follows_type [THEN subsetD])

lemma FollowsD:
"F ∈ Follows(A, r, f, g)==>
F ∈ program & (∃a. a ∈ A) & (∀x ∈ state. f(x):A & g(x):A)"
apply (unfold Follows_def)
apply (blast dest: IncreasingD)
done

lemma Follows_constantI:
"[| F ∈ program; c ∈ A; refl(A, r) |] ==> F ∈ Follows(A, r, %x. c, %x. c)"
apply (unfold Follows_def, auto)
done

lemma subset_Follows_comp:
"[| mono1(A, r, B, s, h); refl(A, r); trans[B](s) |]
==> Follows(A, r, f, g) ⊆ Follows(B, s,  h comp f, h comp g)"
apply (unfold Follows_def, clarify)
apply (frule_tac f = g in IncreasingD)
apply (frule_tac f = f in IncreasingD)
apply (rule IntI)
apply (rule_tac [2] h = h in imp_LeadsTo_comp)
prefer 5 apply assumption
apply (auto intro: imp_Increasing_comp imp_Always_comp simp del: INT_simps)
done

lemma imp_Follows_comp:
"[| F ∈ Follows(A, r, f, g);  mono1(A, r, B, s, h); refl(A, r); trans[B](s) |]
==>  F ∈ Follows(B, s,  h comp f, h comp g)"
apply (blast intro: subset_Follows_comp [THEN subsetD])
done

(* 2-place monotone operation ∈ this general result is used to prove Follows_Un, Follows_munion *)

(* 2-place monotone operation ∈ this general result is
used to prove Follows_Un, Follows_munion *)
lemma imp_Follows_comp2:
"[| F ∈ Follows(A, r, f1, f);  F ∈ Follows(B, s, g1, g);
mono2(A, r, B, s, C, t, h); refl(A,r); refl(B, s); trans[C](t) |]
==> F ∈ Follows(C, t, %x. h(f1(x), g1(x)), %x. h(f(x), g(x)))"
apply (unfold Follows_def, clarify)
apply (frule_tac f = g in IncreasingD)
apply (frule_tac f = f in IncreasingD)
apply (rule IntI, safe)
apply (rule_tac [3] h = h in imp_Always_comp2)
prefer 5 apply assumption
apply (rule_tac [2] h = h in imp_Increasing_comp2)
prefer 4 apply assumption
apply (rule_tac h = h in imp_Increasing_comp2)
prefer 3 apply assumption
apply simp_all
apply (blast dest!: IncreasingD)
apply (rule_tac h = h in imp_LeadsTo_comp2)
prefer 4 apply assumption
apply auto
prefer 3 apply (simp add: mono2_def)
apply (blast dest: IncreasingD)+
done

lemma Follows_trans:
"[| F ∈ Follows(A, r, f, g);  F ∈ Follows(A,r, g, h);
trans[A](r) |] ==> F ∈ Follows(A, r, f, h)"
apply (frule_tac f = f in FollowsD)
apply (frule_tac f = g in FollowsD)
apply (simp add: Always_eq_includes_reachable INT_iff, auto)
apply (rule_tac [2] B = "{s ∈ state. <k, g (s) > ∈ r}" in LeadsTo_Trans)
apply (rule_tac b = "g (x) " in trans_onD)
apply blast+
done

(** Destruction rules for Follows **)

lemma Follows_imp_Increasing_left:
"F ∈ Follows(A, r, f,g) ==> F ∈ Increasing(A, r, f)"
by (unfold Follows_def, blast)

lemma Follows_imp_Increasing_right:
"F ∈ Follows(A, r, f,g) ==> F ∈ Increasing(A, r, g)"
by (unfold Follows_def, blast)

lemma Follows_imp_Always:
"F :Follows(A, r, f, g) ==> F ∈ Always({s ∈ state. <f(s),g(s)> ∈ r})"
by (unfold Follows_def, blast)

"[| F ∈ Follows(A, r, f, g); k ∈ A |]  ==>
F: {s ∈ state. <k,g(s)> ∈ r } ⟼w {s ∈ state. <k,f(s)> ∈ r}"
by (unfold Follows_def, blast)

"[| F ∈ Follows(list(nat), gen_prefix(nat, Le), f, g); k ∈ list(nat) |]
==> F ∈ {s ∈ state. k pfixLe g(s)} ⟼w {s ∈ state. k pfixLe f(s)}"

"[| F ∈ Follows(list(nat), gen_prefix(nat, Ge), f, g); k ∈ list(nat) |]
==> F ∈ {s ∈ state. k pfixGe g(s)} ⟼w {s ∈ state. k pfixGe f(s)}"

lemma Always_Follows1:
"[| F ∈ Always({s ∈ state. f(s) = g(s)}); F ∈ Follows(A, r, f, h);
∀x ∈ state. g(x):A |] ==> F ∈ Follows(A, r, g, h)"
apply (unfold Follows_def Increasing_def Stable_def)
apply (rule_tac [3] C = "{s ∈ state. f(s)=g(s)}"
and A = "{s ∈ state. <k, h (s)> ∈ r}"
and A' = "{s ∈ state. <k, f(s)> ∈ r}" in Always_LeadsTo_weaken)
apply (erule_tac A = "{s ∈ state. <k,f(s) > ∈ r}"
and A' = "{s ∈ state. <k,f(s) > ∈ r}" in Always_Constrains_weaken)
apply auto
apply (drule Always_Int_I, assumption)
apply (erule_tac A = "{s ∈ state. f(s)=g(s)} ∩ {s ∈ state. <f(s), h(s)> ∈ r}"
in Always_weaken)
apply auto
done

lemma Always_Follows2:
"[| F ∈ Always({s ∈ state. g(s) = h(s)});
F ∈ Follows(A, r, f, g); ∀x ∈ state. h(x):A |] ==> F ∈ Follows(A, r, f, h)"
apply (unfold Follows_def Increasing_def Stable_def)
apply (rule_tac [3] C = "{s ∈ state. g (s) =h (s) }"
and A = "{s ∈ state. <k, g (s) > ∈ r}"
and A' = "{s ∈ state. <k, f (s) > ∈ r}" in Always_LeadsTo_weaken)
apply (erule_tac A = "{s ∈ state. <k, g(s)> ∈ r}"
and A' = "{s ∈ state. <k, g(s)> ∈ r}" in Always_Constrains_weaken)
apply auto
apply (drule Always_Int_I, assumption)
apply (erule_tac A = "{s ∈ state. g(s)=h(s)} ∩ {s ∈ state. <f(s), g(s)> ∈ r}"
in Always_weaken)
apply auto
done

(** Union properties (with the subset ordering) **)

lemma refl_SetLe [simp]: "refl(Pow(A), SetLe(A))"
by (unfold refl_def SetLe_def, auto)

lemma trans_on_SetLe [simp]: "trans[Pow(A)](SetLe(A))"
by (unfold trans_on_def SetLe_def, auto)

lemma antisym_SetLe [simp]: "antisym(SetLe(A))"
by (unfold antisym_def SetLe_def, auto)

lemma part_order_SetLe [simp]: "part_order(Pow(A), SetLe(A))"
by (unfold part_order_def, auto)

lemma increasing_Un:
"[| F ∈ Increasing.increasing(Pow(A), SetLe(A), f);
F ∈ Increasing.increasing(Pow(A), SetLe(A), g) |]
==> F ∈ Increasing.increasing(Pow(A), SetLe(A), %x. f(x) ∪ g(x))"
by (rule_tac h = "op Un" in imp_increasing_comp2, auto)

lemma Increasing_Un:
"[| F ∈ Increasing(Pow(A), SetLe(A), f);
F ∈ Increasing(Pow(A), SetLe(A), g) |]
==> F ∈ Increasing(Pow(A), SetLe(A), %x. f(x) ∪ g(x))"
by (rule_tac h = "op Un" in imp_Increasing_comp2, auto)

lemma Always_Un:
"[| F ∈ Always({s ∈ state. f1(s) ⊆ f(s)});
F ∈ Always({s ∈ state. g1(s) ⊆ g(s)}) |]
==> F ∈ Always({s ∈ state. f1(s) ∪ g1(s) ⊆ f(s) ∪ g(s)})"

lemma Follows_Un:
"[| F ∈ Follows(Pow(A), SetLe(A), f1, f);
F ∈ Follows(Pow(A), SetLe(A), g1, g) |]
==> F ∈ Follows(Pow(A), SetLe(A), %s. f1(s) ∪ g1(s), %s. f(s) ∪ g(s))"
by (rule_tac h = "op Un" in imp_Follows_comp2, auto)

(** Multiset union properties (with the MultLe ordering) **)

lemma refl_MultLe [simp]: "refl(Mult(A), MultLe(A,r))"
by (unfold MultLe_def refl_def, auto)

lemma MultLe_refl1 [simp]:
"[| multiset(M); mset_of(M)<=A |] ==> <M, M> ∈ MultLe(A, r)"
apply (unfold MultLe_def id_def lam_def)
done

lemma MultLe_refl2 [simp]: "M ∈ Mult(A) ==> <M, M> ∈ MultLe(A, r)"
by (unfold MultLe_def id_def lam_def, auto)

lemma trans_on_MultLe [simp]: "trans[Mult(A)](MultLe(A,r))"
apply (unfold MultLe_def trans_on_def)
apply (auto intro: trancl_trans simp add: multirel_def)
done

lemma MultLe_type: "MultLe(A, r)<= (Mult(A) * Mult(A))"
apply (unfold MultLe_def, auto)
apply (drule multirel_type [THEN subsetD], auto)
done

lemma MultLe_trans:
"[| <M,K> ∈ MultLe(A,r); <K,N> ∈ MultLe(A,r) |] ==> <M,N> ∈ MultLe(A,r)"
apply (cut_tac A=A in trans_on_MultLe)
apply (drule trans_onD, assumption)
apply (auto dest: MultLe_type [THEN subsetD])
done

lemma part_order_imp_part_ord:
"part_order(A, r) ==> part_ord(A, r-id(A))"
apply (unfold part_order_def part_ord_def)
apply (simp add: refl_def id_def lam_def irrefl_def, auto)
apply auto
apply (blast dest: trans_onD)
apply auto
done

lemma antisym_MultLe [simp]:
"part_order(A, r) ==> antisym(MultLe(A,r))"
apply (unfold MultLe_def antisym_def)
apply (drule part_order_imp_part_ord, auto)
apply (drule irrefl_on_multirel)
apply (frule multirel_type [THEN subsetD])
apply (drule multirel_trans)
done

lemma part_order_MultLe [simp]:
"part_order(A, r) ==>  part_order(Mult(A), MultLe(A, r))"
apply (frule antisym_MultLe)
done

lemma empty_le_MultLe [simp]:
"[| multiset(M); mset_of(M)<= A|] ==> <0, M> ∈ MultLe(A, r)"
apply (unfold MultLe_def)
apply (case_tac "M=0")
apply (subgoal_tac "<0 +# 0, 0 +# M> ∈ multirel (A, r - id (A))")
apply (rule_tac [2] one_step_implies_multirel)
done

lemma empty_le_MultLe2 [simp]: "M ∈ Mult(A) ==> <0, M> ∈ MultLe(A, r)"

lemma munion_mono:
"[| <M, N> ∈ MultLe(A, r); <K, L> ∈ MultLe(A, r) |] ==>
<M +# K, N +# L> ∈ MultLe(A, r)"
apply (unfold MultLe_def)
apply (auto intro: munion_multirel_mono1 munion_multirel_mono2
done

lemma increasing_munion:
"[| F ∈ Increasing.increasing(Mult(A), MultLe(A,r), f);
F ∈ Increasing.increasing(Mult(A), MultLe(A,r), g) |]
==> F ∈ Increasing.increasing(Mult(A),MultLe(A,r), %x. f(x) +# g(x))"
by (rule_tac h = munion in imp_increasing_comp2, auto)

lemma Increasing_munion:
"[| F ∈ Increasing(Mult(A), MultLe(A,r), f);
F ∈ Increasing(Mult(A), MultLe(A,r), g)|]
==> F ∈ Increasing(Mult(A),MultLe(A,r), %x. f(x) +# g(x))"
by (rule_tac h = munion in imp_Increasing_comp2, auto)

lemma Always_munion:
"[| F ∈ Always({s ∈ state. <f1(s),f(s)> ∈ MultLe(A,r)});
F ∈ Always({s ∈ state. <g1(s), g(s)> ∈ MultLe(A,r)});
∀x ∈ state. f1(x):Mult(A)&f(x):Mult(A) & g1(x):Mult(A) & g(x):Mult(A)|]
==> F ∈ Always({s ∈ state. <f1(s) +# g1(s), f(s) +# g(s)> ∈ MultLe(A,r)})"
apply (rule_tac h = munion in imp_Always_comp2, simp_all)
apply (blast intro: munion_mono, simp_all)
done

lemma Follows_munion:
"[| F ∈ Follows(Mult(A), MultLe(A, r), f1, f);
F ∈ Follows(Mult(A), MultLe(A, r), g1, g) |]
==> F ∈ Follows(Mult(A), MultLe(A, r), %s. f1(s) +# g1(s), %s. f(s) +# g(s))"
by (rule_tac h = munion in imp_Follows_comp2, auto)

(** Used in ClientImp **)

lemma Follows_msetsum_UN:
"!!f. [| ∀i ∈ I. F ∈ Follows(Mult(A), MultLe(A, r), f'(i), f(i));
∀s. ∀i ∈ I. multiset(f'(i, s)) & mset_of(f'(i, s))<=A &
multiset(f(i, s)) & mset_of(f(i, s))<=A ;
Finite(I); F ∈ program |]
==> F ∈ Follows(Mult(A),
MultLe(A, r), %x. msetsum(%i. f'(i, x), I, A),
%x. msetsum(%i. f(i,  x), I, A))"
apply (erule rev_mp)
apply (drule Finite_into_Fin)
apply (erule Fin_induct)
apply (simp (no_asm_simp))
apply (rule Follows_constantI)