# Theory Limit

theory Limit
imports ZF
```(*  Title:      ZF/ex/Limit.thy
Author:     Sten Agerholm
Author:     Lawrence C Paulson

A formalization of the inverse limit construction of domain theory.

The following paper comments on the formalization:

"A Comparison of HOL-ST and Isabelle/ZF" by Sten Agerholm
In Proceedings of the First Isabelle Users Workshop, Technical
Report No. 379, University of Cambridge Computer Laboratory, 1995.

This is a condensed version of:

"A Comparison of HOL-ST and Isabelle/ZF" by Sten Agerholm
Technical Report No. 369, University of Cambridge Computer
Laboratory, 1995.
*)

theory Limit  imports  ZF begin

definition
rel :: "[i,i,i]=>o"  where
"rel(D,x,y) == <x,y>:snd(D)"

definition
set :: "i=>i"  where
"set(D) == fst(D)"

definition
po  :: "i=>o"  where
"po(D) ==
(∀x ∈ set(D). rel(D,x,x)) &
(∀x ∈ set(D). ∀y ∈ set(D). ∀z ∈ set(D).
rel(D,x,y) ⟶ rel(D,y,z) ⟶ rel(D,x,z)) &
(∀x ∈ set(D). ∀y ∈ set(D). rel(D,x,y) ⟶ rel(D,y,x) ⟶ x = y)"

definition
chain :: "[i,i]=>o"  where
(* Chains are object level functions nat->set(D) *)
"chain(D,X) == X ∈ nat->set(D) & (∀n ∈ nat. rel(D,X`n,X`(succ(n))))"

definition
isub :: "[i,i,i]=>o"  where
"isub(D,X,x) == x ∈ set(D) & (∀n ∈ nat. rel(D,X`n,x))"

definition
islub :: "[i,i,i]=>o"  where
"islub(D,X,x) == isub(D,X,x) & (∀y. isub(D,X,y) ⟶ rel(D,x,y))"

definition
lub :: "[i,i]=>i"  where
"lub(D,X) == THE x. islub(D,X,x)"

definition
cpo :: "i=>o"  where
"cpo(D) == po(D) & (∀X. chain(D,X) ⟶ (∃x. islub(D,X,x)))"

definition
pcpo :: "i=>o"  where
"pcpo(D) == cpo(D) & (∃x ∈ set(D). ∀y ∈ set(D). rel(D,x,y))"

definition
bot :: "i=>i"  where
"bot(D) == THE x. x ∈ set(D) & (∀y ∈ set(D). rel(D,x,y))"

definition
mono :: "[i,i]=>i"  where
"mono(D,E) ==
{f ∈ set(D)->set(E).
∀x ∈ set(D). ∀y ∈ set(D). rel(D,x,y) ⟶ rel(E,f`x,f`y)}"

definition
cont :: "[i,i]=>i"  where
"cont(D,E) ==
{f ∈ mono(D,E).
∀X. chain(D,X) ⟶ f`(lub(D,X)) = lub(E,λn ∈ nat. f`(X`n))}"

definition
cf :: "[i,i]=>i"  where
"cf(D,E) ==
<cont(D,E),
{y ∈ cont(D,E)*cont(D,E). ∀x ∈ set(D). rel(E,(fst(y))`x,(snd(y))`x)}>"

definition
suffix :: "[i,i]=>i"  where
"suffix(X,n) == λm ∈ nat. X`(n #+ m)"

definition
subchain :: "[i,i]=>o"  where
"subchain(X,Y) == ∀m ∈ nat. ∃n ∈ nat. X`m = Y`(m #+ n)"

definition
dominate :: "[i,i,i]=>o"  where
"dominate(D,X,Y) == ∀m ∈ nat. ∃n ∈ nat. rel(D,X`m,Y`n)"

definition
matrix :: "[i,i]=>o"  where
"matrix(D,M) ==
M ∈ nat -> (nat -> set(D)) &
(∀n ∈ nat. ∀m ∈ nat. rel(D,M`n`m,M`succ(n)`m)) &
(∀n ∈ nat. ∀m ∈ nat. rel(D,M`n`m,M`n`succ(m))) &
(∀n ∈ nat. ∀m ∈ nat. rel(D,M`n`m,M`succ(n)`succ(m)))"

definition
projpair  :: "[i,i,i,i]=>o"  where
"projpair(D,E,e,p) ==
e ∈ cont(D,E) & p ∈ cont(E,D) &
p O e = id(set(D)) & rel(cf(E,E),e O p,id(set(E)))"

definition
emb       :: "[i,i,i]=>o"  where
"emb(D,E,e) == ∃p. projpair(D,E,e,p)"

definition
Rp        :: "[i,i,i]=>i"  where
"Rp(D,E,e) == THE p. projpair(D,E,e,p)"

definition
(* Twice, constructions on cpos are more difficult. *)
iprod     :: "i=>i"  where
"iprod(DD) ==
<(∏n ∈ nat. set(DD`n)),
{x:(∏n ∈ nat. set(DD`n))*(∏n ∈ nat. set(DD`n)).
∀n ∈ nat. rel(DD`n,fst(x)`n,snd(x)`n)}>"

definition
mkcpo     :: "[i,i=>o]=>i"  where
(* Cannot use rel(D), is meta fun, need two more args *)
"mkcpo(D,P) ==
<{x ∈ set(D). P(x)},{x ∈ set(D)*set(D). rel(D,fst(x),snd(x))}>"

definition
subcpo    :: "[i,i]=>o"  where
"subcpo(D,E) ==
set(D) ⊆ set(E) &
(∀x ∈ set(D). ∀y ∈ set(D). rel(D,x,y) ⟷ rel(E,x,y)) &
(∀X. chain(D,X) ⟶ lub(E,X):set(D))"

definition
subpcpo   :: "[i,i]=>o"  where
"subpcpo(D,E) == subcpo(D,E) & bot(E):set(D)"

definition
emb_chain :: "[i,i]=>o"  where
"emb_chain(DD,ee) ==
(∀n ∈ nat. cpo(DD`n)) & (∀n ∈ nat. emb(DD`n,DD`succ(n),ee`n))"

definition
Dinf      :: "[i,i]=>i"  where
"Dinf(DD,ee) ==
mkcpo(iprod(DD))
(%x. ∀n ∈ nat. Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n)"

definition
e_less    :: "[i,i,i,i]=>i"  where
(* Valid for m ≤ n only. *)
"e_less(DD,ee,m,n) == rec(n#-m,id(set(DD`m)),%x y. ee`(m#+x) O y)"

definition
e_gr      :: "[i,i,i,i]=>i"  where
(* Valid for n ≤ m only. *)
"e_gr(DD,ee,m,n) ==
rec(m#-n,id(set(DD`n)),
%x y. y O Rp(DD`(n#+x),DD`(succ(n#+x)),ee`(n#+x)))"

definition
eps       :: "[i,i,i,i]=>i"  where
"eps(DD,ee,m,n) == if(m ≤ n,e_less(DD,ee,m,n),e_gr(DD,ee,m,n))"

definition
rho_emb   :: "[i,i,i]=>i"  where
"rho_emb(DD,ee,n) == λx ∈ set(DD`n). λm ∈ nat. eps(DD,ee,n,m)`x"

definition
rho_proj  :: "[i,i,i]=>i"  where
"rho_proj(DD,ee,n) == λx ∈ set(Dinf(DD,ee)). x`n"

definition
commute   :: "[i,i,i,i=>i]=>o"  where
"commute(DD,ee,E,r) ==
(∀n ∈ nat. emb(DD`n,E,r(n))) &
(∀m ∈ nat. ∀n ∈ nat. m ≤ n ⟶ r(n) O eps(DD,ee,m,n) = r(m))"

definition
mediating :: "[i,i,i=>i,i=>i,i]=>o"  where
"mediating(E,G,r,f,t) == emb(E,G,t) & (∀n ∈ nat. f(n) = t O r(n))"

lemmas nat_linear_le = Ord_linear_le [OF nat_into_Ord nat_into_Ord]

(*----------------------------------------------------------------------*)
(* Basic results.                                                       *)
(*----------------------------------------------------------------------*)

lemma set_I: "x ∈ fst(D) ==> x ∈ set(D)"

lemma rel_I: "<x,y>:snd(D) ==> rel(D,x,y)"

lemma rel_E: "rel(D,x,y) ==> <x,y>:snd(D)"

(*----------------------------------------------------------------------*)
(* I/E/D rules for po and cpo.                                          *)
(*----------------------------------------------------------------------*)

lemma po_refl: "[|po(D); x ∈ set(D)|] ==> rel(D,x,x)"
by (unfold po_def, blast)

lemma po_trans: "[|po(D); rel(D,x,y); rel(D,y,z); x ∈ set(D);
y ∈ set(D); z ∈ set(D)|] ==> rel(D,x,z)"
by (unfold po_def, blast)

lemma po_antisym:
"[|po(D); rel(D,x,y); rel(D,y,x); x ∈ set(D); y ∈ set(D)|] ==> x = y"
by (unfold po_def, blast)

lemma poI:
"[| !!x. x ∈ set(D) ==> rel(D,x,x);
!!x y z. [| rel(D,x,y); rel(D,y,z); x ∈ set(D); y ∈ set(D); z ∈ set(D)|]
==> rel(D,x,z);
!!x y. [| rel(D,x,y); rel(D,y,x); x ∈ set(D); y ∈ set(D)|] ==> x=y |]
==> po(D)"
by (unfold po_def, blast)

lemma cpoI: "[| po(D); !!X. chain(D,X) ==> islub(D,X,x(D,X))|] ==> cpo(D)"

lemma cpo_po: "cpo(D) ==> po(D)"

lemma cpo_refl [simp,intro!,TC]: "[|cpo(D); x ∈ set(D)|] ==> rel(D,x,x)"
by (blast intro: po_refl cpo_po)

lemma cpo_trans:
"[|cpo(D); rel(D,x,y); rel(D,y,z); x ∈ set(D);
y ∈ set(D); z ∈ set(D)|] ==> rel(D,x,z)"
by (blast intro: cpo_po po_trans)

lemma cpo_antisym:
"[|cpo(D); rel(D,x,y); rel(D,y,x); x ∈ set(D); y ∈ set(D)|] ==> x = y"
by (blast intro: cpo_po po_antisym)

lemma cpo_islub: "[|cpo(D); chain(D,X);  !!x. islub(D,X,x) ==> R|] ==> R"

(*----------------------------------------------------------------------*)
(* Theorems about isub and islub.                                       *)
(*----------------------------------------------------------------------*)

lemma islub_isub: "islub(D,X,x) ==> isub(D,X,x)"

lemma islub_in: "islub(D,X,x) ==> x ∈ set(D)"

lemma islub_ub: "[|islub(D,X,x); n ∈ nat|] ==> rel(D,X`n,x)"

lemma islub_least: "[|islub(D,X,x); isub(D,X,y)|] ==> rel(D,x,y)"

lemma islubI:
"[|isub(D,X,x); !!y. isub(D,X,y) ==> rel(D,x,y)|] ==> islub(D,X,x)"

lemma isubI:
"[|x ∈ set(D);  !!n. n ∈ nat ==> rel(D,X`n,x)|] ==> isub(D,X,x)"

lemma isubE:
"[|isub(D,X,x); [|x ∈ set(D);  !!n. n ∈ nat==>rel(D,X`n,x)|] ==> P
|] ==> P"

lemma isubD1: "isub(D,X,x) ==> x ∈ set(D)"

lemma isubD2: "[|isub(D,X,x); n ∈ nat|]==>rel(D,X`n,x)"

lemma islub_unique: "[|islub(D,X,x); islub(D,X,y); cpo(D)|] ==> x = y"
by (blast intro: cpo_antisym islub_least islub_isub islub_in)

(*----------------------------------------------------------------------*)
(* lub gives the least upper bound of chains.                           *)
(*----------------------------------------------------------------------*)

lemma cpo_lub: "[|chain(D,X); cpo(D)|] ==> islub(D,X,lub(D,X))"
apply (best elim: cpo_islub intro: theI islub_unique)
done

(*----------------------------------------------------------------------*)
(*----------------------------------------------------------------------*)

lemma chainI:
"[|X ∈ nat->set(D);  !!n. n ∈ nat ==> rel(D,X`n,X`succ(n))|] ==> chain(D,X)"

lemma chain_fun: "chain(D,X) ==> X ∈ nat -> set(D)"

lemma chain_in [simp,TC]: "[|chain(D,X); n ∈ nat|] ==> X`n ∈ set(D)"
apply (blast dest: apply_type)
done

lemma chain_rel [simp,TC]:
"[|chain(D,X); n ∈ nat|] ==> rel(D, X ` n, X ` succ(n))"

"[|chain(D,X); cpo(D); n ∈ nat; m ∈ nat|] ==> rel(D,X`n,(X`(m #+ n)))"
apply (induct_tac m)
apply (auto intro: cpo_trans)
done

lemma chain_rel_gen:
"[|n ≤ m; chain(D,X); cpo(D); m ∈ nat|] ==> rel(D,X`n,X`m)"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (erule rev_mp) (*prepare the induction*)
apply (induct_tac m)
apply (auto intro: cpo_trans simp add: le_iff)
done

(*----------------------------------------------------------------------*)
(* Theorems about pcpos and bottom.                                     *)
(*----------------------------------------------------------------------*)

lemma pcpoI:
"[|!!y. y ∈ set(D)==>rel(D,x,y); x ∈ set(D); cpo(D)|]==>pcpo(D)"

lemma pcpo_cpo [TC]: "pcpo(D) ==> cpo(D)"

lemma pcpo_bot_ex1:
"pcpo(D) ==> ∃! x. x ∈ set(D) & (∀y ∈ set(D). rel(D,x,y))"
apply (blast intro: cpo_antisym)
done

lemma bot_least [TC]:
"[| pcpo(D); y ∈ set(D)|] ==> rel(D,bot(D),y)"
apply (best intro: pcpo_bot_ex1 [THEN theI2])
done

lemma bot_in [TC]:
"pcpo(D) ==> bot(D):set(D)"
apply (best intro: pcpo_bot_ex1 [THEN theI2])
done

lemma bot_unique:
"[| pcpo(D); x ∈ set(D); !!y. y ∈ set(D) ==> rel(D,x,y)|] ==> x = bot(D)"
by (blast intro: cpo_antisym pcpo_cpo bot_in bot_least)

(*----------------------------------------------------------------------*)
(* Constant chains and lubs and cpos.                                   *)
(*----------------------------------------------------------------------*)

lemma chain_const: "[|x ∈ set(D); cpo(D)|] ==> chain(D,(λn ∈ nat. x))"

lemma islub_const:
"[|x ∈ set(D); cpo(D)|] ==> islub(D,(λn ∈ nat. x),x)"
by (simp add: islub_def isub_def, blast)

lemma lub_const: "[|x ∈ set(D); cpo(D)|] ==> lub(D,λn ∈ nat. x) = x"
by (blast intro: islub_unique cpo_lub chain_const islub_const)

(*----------------------------------------------------------------------*)
(* Taking the suffix of chains has no effect on ub's.                   *)
(*----------------------------------------------------------------------*)

lemma isub_suffix:
"[| chain(D,X); cpo(D) |] ==> isub(D,suffix(X,n),x) ⟷ isub(D,X,x)"
apply (simp add: isub_def suffix_def, safe)
apply (drule_tac x = na in bspec)
done

lemma islub_suffix:
"[|chain(D,X); cpo(D)|] ==> islub(D,suffix(X,n),x) ⟷ islub(D,X,x)"

lemma lub_suffix:
"[|chain(D,X); cpo(D)|] ==> lub(D,suffix(X,n)) = lub(D,X)"

(*----------------------------------------------------------------------*)
(* Dominate and subchain.                                               *)
(*----------------------------------------------------------------------*)

lemma dominateI:
"[| !!m. m ∈ nat ==> n(m):nat; !!m. m ∈ nat ==> rel(D,X`m,Y`n(m))|]
==> dominate(D,X,Y)"

lemma dominate_isub:
"[|dominate(D,X,Y); isub(D,Y,x); cpo(D);
X ∈ nat->set(D); Y ∈ nat->set(D)|] ==> isub(D,X,x)"
apply (blast intro: cpo_trans intro!: apply_funtype)
done

lemma dominate_islub:
"[|dominate(D,X,Y); islub(D,X,x); islub(D,Y,y); cpo(D);
X ∈ nat->set(D); Y ∈ nat->set(D)|] ==> rel(D,x,y)"
apply (blast intro: dominate_isub)
done

lemma subchain_isub:
"[|subchain(Y,X); isub(D,X,x)|] ==> isub(D,Y,x)"
by (simp add: isub_def subchain_def, force)

lemma dominate_islub_eq:
"[|dominate(D,X,Y); subchain(Y,X); islub(D,X,x); islub(D,Y,y); cpo(D);
X ∈ nat->set(D); Y ∈ nat->set(D)|] ==> x = y"
by (blast intro: cpo_antisym dominate_islub islub_least subchain_isub
islub_isub islub_in)

(*----------------------------------------------------------------------*)
(* Matrix.                                                              *)
(*----------------------------------------------------------------------*)

lemma matrix_fun: "matrix(D,M) ==> M ∈ nat -> (nat -> set(D))"

lemma matrix_in_fun: "[|matrix(D,M); n ∈ nat|] ==> M`n ∈ nat -> set(D)"
by (blast intro: apply_funtype matrix_fun)

lemma matrix_in: "[|matrix(D,M); n ∈ nat; m ∈ nat|] ==> M`n`m ∈ set(D)"
by (blast intro: apply_funtype matrix_in_fun)

lemma matrix_rel_1_0:
"[|matrix(D,M); n ∈ nat; m ∈ nat|] ==> rel(D,M`n`m,M`succ(n)`m)"

lemma matrix_rel_0_1:
"[|matrix(D,M); n ∈ nat; m ∈ nat|] ==> rel(D,M`n`m,M`n`succ(m))"

lemma matrix_rel_1_1:
"[|matrix(D,M); n ∈ nat; m ∈ nat|] ==> rel(D,M`n`m,M`succ(n)`succ(m))"

lemma fun_swap: "f ∈ X->Y->Z ==> (λy ∈ Y. λx ∈ X. f`x`y):Y->X->Z"
by (blast intro: lam_type apply_funtype)

lemma matrix_sym_axis:
"matrix(D,M) ==> matrix(D,λm ∈ nat. λn ∈ nat. M`n`m)"

lemma matrix_chain_diag:
"matrix(D,M) ==> chain(D,λn ∈ nat. M`n`n)"
apply (auto intro: lam_type matrix_in matrix_rel_1_1)
done

lemma matrix_chain_left:
"[|matrix(D,M); n ∈ nat|] ==> chain(D,M`n)"
apply (unfold chain_def)
apply (auto intro: matrix_fun [THEN apply_type] matrix_in matrix_rel_0_1)
done

lemma matrix_chain_right:
"[|matrix(D,M); m ∈ nat|] ==> chain(D,λn ∈ nat. M`n`m)"
apply (auto intro: lam_type matrix_in matrix_rel_1_0)
done

lemma matrix_chainI:
assumes xprem: "!!x. x ∈ nat==>chain(D,M`x)"
and yprem: "!!y. y ∈ nat==>chain(D,λx ∈ nat. M`x`y)"
and Mfun:  "M ∈ nat->nat->set(D)"
and cpoD:  "cpo(D)"
shows "matrix(D,M)"
proof -
{
fix n m assume "n ∈ nat" "m ∈ nat"
with chain_rel [OF yprem]
have "rel(D, M ` n ` m, M ` succ(n) ` m)" by simp
} note rel_succ = this
show "matrix(D,M)"
proof (simp add: matrix_def Mfun rel_succ, intro conjI ballI)
fix n m assume n: "n ∈ nat" and m: "m ∈ nat"
thus "rel(D, M ` n ` m, M ` n ` succ(m))"
next
fix n m assume n: "n ∈ nat" and m: "m ∈ nat"
thus "rel(D, M ` n ` m, M ` succ(n) ` succ(m))"
by (rule cpo_trans [OF cpoD rel_succ],
simp_all add: chain_fun [THEN apply_type] xprem)
qed
qed

lemma lemma2:
"[|x ∈ nat; m ∈ nat; rel(D,(λn ∈ nat. M`n`m1)`x,(λn ∈ nat. M`n`m1)`m)|]
==> rel(D,M`x`m1,M`m`m1)"
by simp

lemma isub_lemma:
"[|isub(D, λn ∈ nat. M`n`n, y); matrix(D,M); cpo(D)|]
==> isub(D, λn ∈ nat. lub(D,λm ∈ nat. M`n`m), y)"
fix n
assume DM: "matrix(D, M)" and D: "cpo(D)" and n: "n ∈ nat" and y: "y ∈ set(D)"
and rel: "∀n∈nat. rel(D, M ` n ` n, y)"
have "rel(D, lub(D, M ` n), y)"
proof (rule matrix_chain_left [THEN cpo_lub, THEN islub_least], simp_all add: n D DM)
show "isub(D, M ` n, y)"
proof (unfold isub_def, intro conjI ballI y)
fix k assume k: "k ∈ nat"
show "rel(D, M ` n ` k, y)"
proof (cases "n ≤ k")
case True
hence yy: "rel(D, M`n`k, M`k`k)"
by (blast intro: lemma2 n k y DM D chain_rel_gen matrix_chain_right)
show "?thesis"
by (rule cpo_trans [OF D yy],
simp_all add: k rel n y DM matrix_in)
next
case False
hence le: "k ≤ n"
by (blast intro: not_le_iff_lt [THEN iffD1, THEN leI] nat_into_Ord n k)
show "?thesis"
by (rule cpo_trans [OF D chain_rel_gen [OF le]],
simp_all add: n y k rel DM D matrix_chain_left)
qed
qed
qed
moreover
have "M ` n ∈ nat → set(D)" by (blast intro: DM n matrix_fun [THEN apply_type])
ultimately show "rel(D, lub(D, Lambda(nat, op `(M ` n))), y)"  by simp
qed

lemma matrix_chain_lub:
"[|matrix(D,M); cpo(D)|] ==> chain(D,λn ∈ nat. lub(D,λm ∈ nat. M`n`m))"
proof (simp add: chain_def, intro conjI ballI)
assume "matrix(D, M)" "cpo(D)"
thus "(λx∈nat. lub(D, Lambda(nat, op `(M ` x)))) ∈ nat → set(D)"
by (force intro: islub_in cpo_lub chainI lam_type matrix_in matrix_rel_0_1)
next
fix n
assume DD: "matrix(D, M)" "cpo(D)" "n ∈ nat"
hence "dominate(D, M ` n, M ` succ(n))"
by (force simp add: dominate_def intro: matrix_rel_1_0)
with DD show "rel(D, lub(D, Lambda(nat, op `(M ` n))),
lub(D, Lambda(nat, op `(M ` succ(n)))))"
by (simp add: matrix_chain_left [THEN chain_fun, THEN eta]
dominate_islub cpo_lub matrix_chain_left chain_fun)
qed

lemma isub_eq:
assumes DM: "matrix(D, M)" and D: "cpo(D)"
shows "isub(D,(λn ∈ nat. lub(D,λm ∈ nat. M`n`m)),y) ⟷ isub(D,(λn ∈ nat. M`n`n),y)"
proof
assume isub: "isub(D, λn∈nat. lub(D, Lambda(nat, op `(M ` n))), y)"
hence dom: "dominate(D, λn∈nat. M ` n ` n, λn∈nat. lub(D, Lambda(nat, op `(M ` n))))"
using DM D
by (simp add: dominate_def, intro ballI bexI,
simp_all add: matrix_chain_left [THEN chain_fun, THEN eta] islub_ub cpo_lub matrix_chain_left)
thus "isub(D, λn∈nat. M ` n ` n, y)" using DM D
by - (rule dominate_isub [OF dom isub],
next
assume isub: "isub(D, λn∈nat. M ` n ` n, y)"
thus "isub(D, λn∈nat. lub(D, Lambda(nat, op `(M ` n))), y)"  using DM D
qed

lemma lub_matrix_diag_aux1:
"lub(D,(λn ∈ nat. lub(D,λm ∈ nat. M`n`m))) =
(THE x. islub(D, (λn ∈ nat. lub(D,λm ∈ nat. M`n`m)), x))"

lemma lub_matrix_diag_aux2:
"lub(D,(λn ∈ nat. M`n`n)) =
(THE x. islub(D, (λn ∈ nat. M`n`n), x))"

lemma lub_matrix_diag:
"[|matrix(D,M); cpo(D)|]
==> lub(D,(λn ∈ nat. lub(D,λm ∈ nat. M`n`m))) =
lub(D,(λn ∈ nat. M`n`n))"
apply (simp (no_asm) add: lub_matrix_diag_aux1 lub_matrix_diag_aux2)
done

lemma lub_matrix_diag_sym:
"[|matrix(D,M); cpo(D)|]
==> lub(D,(λm ∈ nat. lub(D,λn ∈ nat. M`n`m))) =
lub(D,(λn ∈ nat. M`n`n))"
by (drule matrix_sym_axis [THEN lub_matrix_diag], auto)

(*----------------------------------------------------------------------*)
(* I/E/D rules for mono and cont.                                       *)
(*----------------------------------------------------------------------*)

lemma monoI:
"[|f ∈ set(D)->set(E);
!!x y. [|rel(D,x,y); x ∈ set(D); y ∈ set(D)|] ==> rel(E,f`x,f`y)|]
==> f ∈ mono(D,E)"

lemma mono_fun: "f ∈ mono(D,E) ==> f ∈ set(D)->set(E)"

lemma mono_map: "[|f ∈ mono(D,E); x ∈ set(D)|] ==> f`x ∈ set(E)"
by (blast intro!: mono_fun [THEN apply_type])

lemma mono_mono:
"[|f ∈ mono(D,E); rel(D,x,y); x ∈ set(D); y ∈ set(D)|] ==> rel(E,f`x,f`y)"

lemma contI:
"[|f ∈ set(D)->set(E);
!!x y. [|rel(D,x,y); x ∈ set(D); y ∈ set(D)|] ==> rel(E,f`x,f`y);
!!X. chain(D,X) ==> f`lub(D,X) = lub(E,λn ∈ nat. f`(X`n))|]
==> f ∈ cont(D,E)"

lemma cont2mono: "f ∈ cont(D,E) ==> f ∈ mono(D,E)"

lemma cont_fun [TC]: "f ∈ cont(D,E) ==> f ∈ set(D)->set(E)"
apply (rule mono_fun, blast)
done

lemma cont_map [TC]: "[|f ∈ cont(D,E); x ∈ set(D)|] ==> f`x ∈ set(E)"
by (blast intro!: cont_fun [THEN apply_type])

declare comp_fun [TC]

lemma cont_mono:
"[|f ∈ cont(D,E); rel(D,x,y); x ∈ set(D); y ∈ set(D)|] ==> rel(E,f`x,f`y)"
apply (blast intro!: mono_mono)
done

lemma cont_lub:
"[|f ∈ cont(D,E); chain(D,X)|] ==> f`(lub(D,X)) = lub(E,λn ∈ nat. f`(X`n))"

(*----------------------------------------------------------------------*)
(* Continuity and chains.                                               *)
(*----------------------------------------------------------------------*)

lemma mono_chain:
"[|f ∈ mono(D,E); chain(D,X)|] ==> chain(E,λn ∈ nat. f`(X`n))"
apply (blast intro: lam_type mono_map chain_in mono_mono chain_rel)
done

lemma cont_chain:
"[|f ∈ cont(D,E); chain(D,X)|] ==> chain(E,λn ∈ nat. f`(X`n))"
by (blast intro: mono_chain cont2mono)

(*----------------------------------------------------------------------*)
(* I/E/D rules about (set+rel) cf, the continuous function space.       *)
(*----------------------------------------------------------------------*)

(* The following development more difficult with cpo-as-relation approach. *)

lemma cf_cont: "f ∈ set(cf(D,E)) ==> f ∈ cont(D,E)"

lemma cont_cf: (* Non-trivial with relation *)
"f ∈ cont(D,E) ==> f ∈ set(cf(D,E))"

(* rel_cf originally an equality. Now stated as two rules. Seemed easiest. *)

lemma rel_cfI:
"[|!!x. x ∈ set(D) ==> rel(E,f`x,g`x); f ∈ cont(D,E); g ∈ cont(D,E)|]
==> rel(cf(D,E),f,g)"

lemma rel_cf: "[|rel(cf(D,E),f,g); x ∈ set(D)|] ==> rel(E,f`x,g`x)"

(*----------------------------------------------------------------------*)
(* Theorems about the continuous function space.                        *)
(*----------------------------------------------------------------------*)

lemma chain_cf:
"[| chain(cf(D,E),X); x ∈ set(D)|] ==> chain(E,λn ∈ nat. X`n`x)"
apply (rule chainI)
apply (blast intro: lam_type apply_funtype cont_fun cf_cont chain_in, simp)
apply (blast intro: rel_cf chain_rel)
done

lemma matrix_lemma:
"[|chain(cf(D,E),X); chain(D,Xa); cpo(D); cpo(E) |]
==> matrix(E,λx ∈ nat. λxa ∈ nat. X`x`(Xa`xa))"
apply (rule matrix_chainI, auto)
apply (force intro: chainI lam_type apply_funtype cont_fun cf_cont cont_mono)
apply (force intro: chainI lam_type apply_funtype cont_fun cf_cont rel_cf)
apply (blast intro: lam_type apply_funtype cont_fun cf_cont chain_in)
done

lemma chain_cf_lub_cont:
assumes ch: "chain(cf(D,E),X)" and D: "cpo(D)" and E: "cpo(E)"
shows   "(λx ∈ set(D). lub(E, λn ∈ nat. X ` n ` x)) ∈ cont(D, E)"
proof (rule contI)
show "(λx∈set(D). lub(E, λn∈nat. X ` n ` x)) ∈ set(D) → set(E)"
by (blast intro: lam_type chain_cf [THEN cpo_lub, THEN islub_in] ch E)
next
fix x y
assume xy: "rel(D, x, y)" "x ∈ set(D)" "y ∈ set(D)"
hence dom: "dominate(E, λn∈nat. X ` n ` x, λn∈nat. X ` n ` y)"
by (force intro: dominateI  chain_in [OF ch, THEN cf_cont, THEN cont_mono])
note chE = chain_cf [OF ch]
from xy show "rel(E, (λx∈set(D). lub(E, λn∈nat. X ` n ` x)) ` x,
(λx∈set(D). lub(E, λn∈nat. X ` n ` x)) ` y)"
by (simp add: dominate_islub [OF dom] cpo_lub [OF chE] E chain_fun [OF chE])
next
fix Y
assume chDY: "chain(D,Y)"
have "lub(E, λx∈nat. lub(E, λy∈nat. X ` x ` (Y ` y))) =
lub(E, λx∈nat. X ` x ` (Y ` x))"
using matrix_lemma [THEN lub_matrix_diag, OF ch chDY]
also have "... =  lub(E, λx∈nat. lub(E, λn∈nat. X ` n ` (Y ` x)))"
using  matrix_lemma [THEN lub_matrix_diag_sym, OF ch chDY]
finally have "lub(E, λx∈nat. lub(E, λn∈nat. X ` x ` (Y ` n))) =
lub(E, λx∈nat. lub(E, λn∈nat. X ` n ` (Y ` x)))" .
thus "(λx∈set(D). lub(E, λn∈nat. X ` n ` x)) ` lub(D, Y) =
lub(E, λn∈nat. (λx∈set(D). lub(E, λn∈nat. X ` n ` x)) ` (Y ` n))"
by (simp add: cpo_lub [THEN islub_in]  D chDY
chain_in [THEN cf_cont, THEN cont_lub, OF ch])
qed

lemma islub_cf:
"[| chain(cf(D,E),X); cpo(D); cpo(E)|]
==> islub(cf(D,E), X, λx ∈ set(D). lub(E,λn ∈ nat. X`n`x))"
apply (rule islubI)
apply (rule isubI)
apply (rule chain_cf_lub_cont [THEN cont_cf], assumption+)
apply (rule rel_cfI)
apply (force dest!: chain_cf [THEN cpo_lub, THEN islub_ub])
apply (blast intro: cf_cont chain_in)
apply (blast intro: cont_cf chain_cf_lub_cont)
apply (rule rel_cfI, simp)
apply (force intro: chain_cf [THEN cpo_lub, THEN islub_least]
cf_cont [THEN cont_fun, THEN apply_type] isubI
elim: isubD2 [THEN rel_cf] isubD1)
apply (blast intro: chain_cf_lub_cont isubD1 cf_cont)+
done

lemma cpo_cf [TC]:
"[| cpo(D); cpo(E)|] ==> cpo(cf(D,E))"
apply (rule poI [THEN cpoI])
apply (rule rel_cfI)
apply (assumption | rule cpo_refl cf_cont [THEN cont_fun, THEN apply_type]
cf_cont)+
apply (rule rel_cfI)
apply (rule cpo_trans, assumption)
apply (erule rel_cf, assumption)
apply (rule rel_cf, assumption)
apply (assumption | rule cf_cont [THEN cont_fun, THEN apply_type] cf_cont)+
apply (rule fun_extension)
apply (assumption | rule cf_cont [THEN cont_fun])+
apply (blast intro: cpo_antisym rel_cf
cf_cont [THEN cont_fun, THEN apply_type])
apply (fast intro: islub_cf)
done

lemma lub_cf:
"[| chain(cf(D,E),X); cpo(D); cpo(E)|]
==> lub(cf(D,E), X) = (λx ∈ set(D). lub(E,λn ∈ nat. X`n`x))"
by (blast intro: islub_unique cpo_lub islub_cf cpo_cf)

lemma const_cont [TC]:
"[|y ∈ set(E); cpo(D); cpo(E)|] ==> (λx ∈ set(D).y) ∈ cont(D,E)"
apply (rule contI)
prefer 2 apply simp
apply (blast intro: lam_type)
apply (simp add: chain_in cpo_lub [THEN islub_in] lub_const)
done

lemma cf_least:
"[|cpo(D); pcpo(E); y ∈ cont(D,E)|]==>rel(cf(D,E),(λx ∈ set(D).bot(E)),y)"
apply (rule rel_cfI, simp, typecheck)
done

lemma pcpo_cf:
"[|cpo(D); pcpo(E)|] ==> pcpo(cf(D,E))"
apply (rule pcpoI)
apply (assumption |
rule cf_least bot_in const_cont [THEN cont_cf] cf_cont cpo_cf pcpo_cpo)+
done

lemma bot_cf:
"[|cpo(D); pcpo(E)|] ==> bot(cf(D,E)) = (λx ∈ set(D).bot(E))"
by (blast intro: bot_unique [symmetric] pcpo_cf cf_least
bot_in [THEN const_cont, THEN cont_cf] cf_cont pcpo_cpo)

(*----------------------------------------------------------------------*)
(* Identity and composition.                                            *)
(*----------------------------------------------------------------------*)

lemma id_cont [TC,intro!]:
"cpo(D) ==> id(set(D)) ∈ cont(D,D)"
by (simp add: id_type contI cpo_lub [THEN islub_in] chain_fun [THEN eta])

lemmas comp_cont_apply =  cont_fun [THEN comp_fun_apply]

lemma comp_pres_cont [TC]:
"[| f ∈ cont(D',E); g ∈ cont(D,D'); cpo(D)|] ==> f O g ∈ cont(D,E)"
apply (rule contI)
apply (rule_tac [2] comp_cont_apply [THEN ssubst])
apply (rule_tac [4] comp_cont_apply [THEN ssubst])
apply (rule_tac [6] cont_mono)
apply (rule_tac [7] cont_mono) (* 13 subgoals *)
apply typecheck (* proves all but the lub case *)
apply (subst comp_cont_apply)
apply (rule_tac [3] cont_lub [THEN ssubst])
apply (rule_tac [5] cont_lub [THEN ssubst])
prefer 7 apply (simp add: comp_cont_apply chain_in)
apply (auto intro: cpo_lub [THEN islub_in] cont_chain)
done

lemma comp_mono:
"[| f ∈ cont(D',E); g ∈ cont(D,D'); f':cont(D',E); g':cont(D,D');
rel(cf(D',E),f,f'); rel(cf(D,D'),g,g'); cpo(D); cpo(E) |]
==> rel(cf(D,E),f O g,f' O g')"
apply (rule rel_cfI)
apply (subst comp_cont_apply)
apply (rule_tac [3] comp_cont_apply [THEN ssubst])
apply (rule_tac [5] cpo_trans)
apply (assumption | rule rel_cf cont_mono cont_map comp_pres_cont)+
done

lemma chain_cf_comp:
"[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(E)|]
==> chain(cf(D,E),λn ∈ nat. X`n O Y`n)"
apply (rule chainI)
defer 1
apply simp
apply (rule rel_cfI)
apply (rule comp_cont_apply [THEN ssubst])
apply (rule_tac [3] comp_cont_apply [THEN ssubst])
apply (rule_tac [5] cpo_trans)
apply (rule_tac [6] rel_cf)
apply (rule_tac [8] cont_mono)
apply (blast intro: lam_type comp_pres_cont cont_cf chain_in [THEN cf_cont]
cont_map chain_rel rel_cf)+
done

lemma comp_lubs:
"[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(D'); cpo(E)|]
==> lub(cf(D',E),X) O lub(cf(D,D'),Y) = lub(cf(D,E),λn ∈ nat. X`n O Y`n)"
apply (rule fun_extension)
apply (rule_tac [3] lub_cf [THEN ssubst])
apply (assumption |
rule comp_fun cf_cont [THEN cont_fun]  cpo_lub [THEN islub_in]
cpo_cf chain_cf_comp)+
apply (simp add: chain_in [THEN cf_cont, THEN comp_cont_apply])
apply (subst comp_cont_apply)
apply (assumption | rule cpo_lub [THEN islub_in, THEN cf_cont]  cpo_cf)+
apply (simp add: lub_cf chain_cf chain_in [THEN cf_cont, THEN cont_lub]
chain_cf [THEN cpo_lub, THEN islub_in])
apply (cut_tac M = "λxa ∈ nat. λxb ∈ nat. X`xa` (Y`xb`x)" in lub_matrix_diag)
prefer 3 apply simp
apply (rule matrix_chainI, simp_all)
apply (drule chain_in [THEN cf_cont], assumption)
apply (force dest: cont_chain [OF _ chain_cf])
apply (rule chain_cf)
apply (assumption |
rule cont_fun [THEN apply_type] chain_in [THEN cf_cont] lam_type)+
done

(*----------------------------------------------------------------------*)
(*----------------------------------------------------------------------*)

lemma projpairI:
"[| e ∈ cont(D,E); p ∈ cont(E,D); p O e = id(set(D));
rel(cf(E,E))(e O p)(id(set(E)))|] ==> projpair(D,E,e,p)"

lemma projpair_e_cont: "projpair(D,E,e,p) ==> e ∈ cont(D,E)"

lemma projpair_p_cont: "projpair(D,E,e,p) ==> p ∈ cont(E,D)"

lemma projpair_ep_cont: "projpair(D,E,e,p) ==> e ∈ cont(D,E) & p ∈ cont(E,D)"

lemma projpair_eq: "projpair(D,E,e,p) ==> p O e = id(set(D))"

lemma projpair_rel: "projpair(D,E,e,p) ==> rel(cf(E,E))(e O p)(id(set(E)))"

(*----------------------------------------------------------------------*)
(* NB! projpair_e_cont and projpair_p_cont cannot be used repeatedly    *)
(*   at the same time since both match a goal of the form f ∈ cont(X,Y).*)
(*----------------------------------------------------------------------*)

(*----------------------------------------------------------------------*)
(* Uniqueness of embedding projection pairs.                            *)
(*----------------------------------------------------------------------*)

lemmas id_comp = fun_is_rel [THEN left_comp_id]
and    comp_id = fun_is_rel [THEN right_comp_id]

lemma projpair_unique_aux1:
"[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p');
rel(cf(D,E),e,e')|] ==> rel(cf(E,D),p',p)"
apply (rule_tac b=p' in
projpair_p_cont [THEN cont_fun, THEN id_comp, THEN subst], assumption)
apply (rule projpair_eq [THEN subst], assumption)
apply (rule cpo_trans)
apply (assumption | rule cpo_cf)+
(* The following corresponds to EXISTS_TAC, non-trivial instantiation. *)
apply (rule_tac [4] f = "p O (e' O p')" in cont_cf)
apply (subst comp_assoc)
apply (blast intro:  cpo_cf cont_cf comp_mono comp_pres_cont
dest: projpair_ep_cont)
apply (rule_tac P = "%x. rel (cf (E,D),p O e' O p',x)"
in projpair_p_cont [THEN cont_fun, THEN comp_id, THEN subst],
assumption)
apply (rule comp_mono)
apply (blast intro: cpo_cf cont_cf comp_pres_cont projpair_rel
dest: projpair_ep_cont)+
done

text‹Proof's very like the previous one.  Is there a pattern that
could be exploited?›
lemma projpair_unique_aux2:
"[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p');
rel(cf(E,D),p',p)|] ==> rel(cf(D,E),e,e')"
apply (rule_tac b=e
in projpair_e_cont [THEN cont_fun, THEN comp_id, THEN subst],
assumption)
apply (rule_tac e1=e' in projpair_eq [THEN subst], assumption)
apply (rule cpo_trans)
apply (assumption | rule cpo_cf)+
apply (rule_tac [4] f = "(e O p) O e'" in cont_cf)
apply (subst comp_assoc)
apply (blast intro:  cpo_cf cont_cf comp_mono comp_pres_cont
dest: projpair_ep_cont)
apply (rule_tac P = "%x. rel (cf (D,E), (e O p) O e',x)"
in projpair_e_cont [THEN cont_fun, THEN id_comp, THEN subst],
assumption)
apply (blast intro: cpo_cf cont_cf comp_pres_cont projpair_rel comp_mono
dest: projpair_ep_cont)+
done

lemma projpair_unique:
"[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p')|]
==> (e=e')⟷(p=p')"
by (blast intro: cpo_antisym projpair_unique_aux1 projpair_unique_aux2 cpo_cf cont_cf
dest: projpair_ep_cont)

(* Slightly different, more asms, since THE chooses the unique element. *)

lemma embRp:
"[|emb(D,E,e); cpo(D); cpo(E)|] ==> projpair(D,E,e,Rp(D,E,e))"
apply (blast intro: theI2 projpair_unique [THEN iffD1])
done

lemma embI: "projpair(D,E,e,p) ==> emb(D,E,e)"

lemma Rp_unique: "[|projpair(D,E,e,p); cpo(D); cpo(E)|] ==> Rp(D,E,e) = p"
by (blast intro: embRp embI projpair_unique [THEN iffD1])

lemma emb_cont [TC]: "emb(D,E,e) ==> e ∈ cont(D,E)"
apply (blast intro: projpair_e_cont)
done

(* The following three theorems have cpo asms due to THE (uniqueness). *)

lemmas Rp_cont [TC] = embRp [THEN projpair_p_cont]
lemmas embRp_eq = embRp [THEN projpair_eq]
lemmas embRp_rel = embRp [THEN projpair_rel]

lemma embRp_eq_thm:
"[|emb(D,E,e); x ∈ set(D); cpo(D); cpo(E)|] ==> Rp(D,E,e)`(e`x) = x"
apply (rule comp_fun_apply [THEN subst])
apply (assumption | rule Rp_cont emb_cont cont_fun)+
apply (subst embRp_eq)
apply (auto intro: id_conv)
done

(*----------------------------------------------------------------------*)
(* The identity embedding.                                              *)
(*----------------------------------------------------------------------*)

lemma projpair_id:
"cpo(D) ==> projpair(D,D,id(set(D)),id(set(D)))"
apply (blast intro: cpo_cf cont_cf)
done

lemma emb_id:
"cpo(D) ==> emb(D,D,id(set(D)))"
by (auto intro: embI projpair_id)

lemma Rp_id:
"cpo(D) ==> Rp(D,D,id(set(D))) = id(set(D))"
by (auto intro: Rp_unique projpair_id)

(*----------------------------------------------------------------------*)
(* Composition preserves embeddings.                                    *)
(*----------------------------------------------------------------------*)

(* Considerably shorter, only partly due to a simpler comp_assoc. *)
(* Proof in HOL-ST: 70 lines (minus 14 due to comp_assoc complication). *)
(* Proof in Isa/ZF: 23 lines (compared to 56: 60% reduction). *)

lemma comp_lemma:
"[|emb(D,D',e); emb(D',E,e'); cpo(D); cpo(D'); cpo(E)|]
==> projpair(D,E,e' O e,(Rp(D,D',e)) O (Rp(D',E,e')))"
apply (assumption | rule comp_pres_cont Rp_cont emb_cont)+
apply (rule comp_assoc [THEN subst])
apply (rule_tac t1 = e' in comp_assoc [THEN ssubst])
apply (subst embRp_eq) (* Matches everything due to subst/ssubst. *)
apply assumption+
apply (subst comp_id)
apply (assumption | rule cont_fun Rp_cont embRp_eq)+
apply (rule comp_assoc [THEN subst])
apply (rule_tac t1 = "Rp (D,D',e)" in comp_assoc [THEN ssubst])
apply (rule cpo_trans)
apply (assumption | rule cpo_cf)+
apply (rule comp_mono)
apply (rule_tac [6] cpo_refl)
apply (erule_tac [7] asm_rl | rule_tac [7] cont_cf Rp_cont)+
prefer 6 apply (blast intro: cpo_cf)
apply (rule_tac [5] comp_mono)
apply (rule_tac [10] embRp_rel)
apply (rule_tac [9] cpo_cf [THEN cpo_refl])
apply (simp_all add: comp_id embRp_rel comp_pres_cont Rp_cont
id_cont emb_cont cont_fun cont_cf)
done

(* The use of THEN is great in places like the following, both ugly in HOL. *)

lemmas emb_comp = comp_lemma [THEN embI]
lemmas Rp_comp = comp_lemma [THEN Rp_unique]

(*----------------------------------------------------------------------*)
(* Infinite cartesian product.                                          *)
(*----------------------------------------------------------------------*)

lemma iprodI:
"x:(∏n ∈ nat. set(DD`n)) ==> x ∈ set(iprod(DD))"

lemma iprodE:
"x ∈ set(iprod(DD)) ==> x:(∏n ∈ nat. set(DD`n))"

(* Contains typing conditions in contrast to HOL-ST *)

lemma rel_iprodI:
"[|!!n. n ∈ nat ==> rel(DD`n,f`n,g`n); f:(∏n ∈ nat. set(DD`n));
g:(∏n ∈ nat. set(DD`n))|] ==> rel(iprod(DD),f,g)"

lemma rel_iprodE:
"[|rel(iprod(DD),f,g); n ∈ nat|] ==> rel(DD`n,f`n,g`n)"

lemma chain_iprod:
"[|chain(iprod(DD),X);  !!n. n ∈ nat ==> cpo(DD`n); n ∈ nat|]
==> chain(DD`n,λm ∈ nat. X`m`n)"
apply (unfold chain_def, safe)
apply (rule lam_type)
apply (rule apply_type)
apply (rule iprodE)
apply (blast intro: apply_funtype, assumption)
done

lemma islub_iprod:
"[|chain(iprod(DD),X);  !!n. n ∈ nat ==> cpo(DD`n)|]
==> islub(iprod(DD),X,λn ∈ nat. lub(DD`n,λm ∈ nat. X`m`n))"
apply (simp add: islub_def isub_def, safe)
apply (rule iprodI)
apply (blast intro: lam_type chain_iprod [THEN cpo_lub, THEN islub_in])
apply (rule rel_iprodI, simp)
(*looks like something should be inserted into the assumptions!*)
apply (rule_tac P = "%t. rel (DD`na,t,lub (DD`na,λx ∈ nat. X`x`na))"
and b1 = "%n. X`n`na" in beta [THEN subst])
apply (simp del: beta_if
add: chain_iprod [THEN cpo_lub, THEN islub_ub] iprodE
chain_in)+
apply (blast intro: iprodI lam_type chain_iprod [THEN cpo_lub, THEN islub_in])
apply (rule rel_iprodI)
apply (simp | rule islub_least chain_iprod [THEN cpo_lub])+
apply (erule iprodE [THEN apply_type])
apply (simp_all add: rel_iprodE lam_type iprodE
chain_iprod [THEN cpo_lub, THEN islub_in])
done

lemma cpo_iprod [TC]:
"(!!n. n ∈ nat ==> cpo(DD`n)) ==> cpo(iprod(DD))"
apply (assumption | rule cpoI poI)+
apply (rule rel_iprodI) (*not repeated: want to solve 1, leave 2 unchanged *)
apply (simp | rule cpo_refl iprodE [THEN apply_type] iprodE)+
apply (rule rel_iprodI)
apply (drule rel_iprodE)
apply (drule_tac [2] rel_iprodE)
apply (simp | rule cpo_trans iprodE [THEN apply_type] iprodE)+
apply (rule fun_extension)
apply (blast intro: iprodE)
apply (blast intro: iprodE)
apply (blast intro: cpo_antisym rel_iprodE iprodE [THEN apply_type])+
apply (auto intro: islub_iprod)
done

lemma lub_iprod:
"[|chain(iprod(DD),X);  !!n. n ∈ nat ==> cpo(DD`n)|]
==> lub(iprod(DD),X) = (λn ∈ nat. lub(DD`n,λm ∈ nat. X`m`n))"
by (blast intro: cpo_lub [THEN islub_unique] islub_iprod cpo_iprod)

(*----------------------------------------------------------------------*)
(* The notion of subcpo.                                                *)
(*----------------------------------------------------------------------*)

lemma subcpoI:
"[|set(D)<=set(E);
!!x y. [|x ∈ set(D); y ∈ set(D)|] ==> rel(D,x,y)⟷rel(E,x,y);
!!X. chain(D,X) ==> lub(E,X) ∈ set(D)|] ==> subcpo(D,E)"

lemma subcpo_subset: "subcpo(D,E) ==> set(D)<=set(E)"

lemma subcpo_rel_eq:
"[|subcpo(D,E); x ∈ set(D); y ∈ set(D)|] ==> rel(D,x,y)⟷rel(E,x,y)"

lemmas subcpo_relD1 = subcpo_rel_eq [THEN iffD1]
lemmas subcpo_relD2 = subcpo_rel_eq [THEN iffD2]

lemma subcpo_lub: "[|subcpo(D,E); chain(D,X)|] ==> lub(E,X) ∈ set(D)"

lemma chain_subcpo: "[|subcpo(D,E); chain(D,X)|] ==> chain(E,X)"
by (blast intro: Pi_type [THEN chainI] chain_fun subcpo_relD1
subcpo_subset [THEN subsetD]
chain_in chain_rel)

lemma ub_subcpo: "[|subcpo(D,E); chain(D,X); isub(D,X,x)|] ==> isub(E,X,x)"
by (blast intro: isubI subcpo_relD1 subcpo_relD1 chain_in isubD1 isubD2
subcpo_subset [THEN subsetD] chain_in chain_rel)

lemma islub_subcpo:
"[|subcpo(D,E); cpo(E); chain(D,X)|] ==> islub(D,X,lub(E,X))"
by (blast intro: islubI isubI subcpo_lub subcpo_relD2 chain_in islub_ub
islub_least cpo_lub chain_subcpo isubD1 ub_subcpo)

lemma subcpo_cpo: "[|subcpo(D,E); cpo(E)|] ==> cpo(D)"
apply (assumption | rule cpoI poI)+
apply (assumption | rule cpo_refl subcpo_subset [THEN subsetD])+
apply (blast intro: subcpo_subset [THEN subsetD] cpo_trans)
apply (blast intro: cpo_antisym subcpo_subset [THEN subsetD])
apply (fast intro: islub_subcpo)
done

lemma lub_subcpo: "[|subcpo(D,E); cpo(E); chain(D,X)|] ==> lub(D,X) = lub(E,X)"
by (blast intro: cpo_lub [THEN islub_unique] islub_subcpo subcpo_cpo)

(*----------------------------------------------------------------------*)
(* Making subcpos using mkcpo.                                          *)
(*----------------------------------------------------------------------*)

lemma mkcpoI: "[|x ∈ set(D); P(x)|] ==> x ∈ set(mkcpo(D,P))"

lemma mkcpoD1: "x ∈ set(mkcpo(D,P))==> x ∈ set(D)"

lemma mkcpoD2: "x ∈ set(mkcpo(D,P))==> P(x)"

lemma rel_mkcpoE: "rel(mkcpo(D,P),x,y) ==> rel(D,x,y)"

lemma rel_mkcpo:
"[|x ∈ set(D); y ∈ set(D)|] ==> rel(mkcpo(D,P),x,y) ⟷ rel(D,x,y)"
by (simp add: mkcpo_def rel_def set_def)

lemma chain_mkcpo:
"chain(mkcpo(D,P),X) ==> chain(D,X)"
apply (rule chainI)
apply (blast intro: Pi_type chain_fun chain_in [THEN mkcpoD1])
apply (blast intro: rel_mkcpo [THEN iffD1] chain_rel mkcpoD1 chain_in)
done

lemma subcpo_mkcpo:
"[|!!X. chain(mkcpo(D,P),X) ==> P(lub(D,X)); cpo(D)|]
==> subcpo(mkcpo(D,P),D)"
apply (intro subcpoI subsetI rel_mkcpo)
apply (erule mkcpoD1)+
apply (blast intro: mkcpoI cpo_lub [THEN islub_in] chain_mkcpo)
done

(*----------------------------------------------------------------------*)
(* Embedding projection chains of cpos.                                 *)
(*----------------------------------------------------------------------*)

lemma emb_chainI:
"[|!!n. n ∈ nat ==> cpo(DD`n);
!!n. n ∈ nat ==> emb(DD`n,DD`succ(n),ee`n)|] ==> emb_chain(DD,ee)"

lemma emb_chain_cpo [TC]: "[|emb_chain(DD,ee); n ∈ nat|] ==> cpo(DD`n)"

lemma emb_chain_emb:
"[|emb_chain(DD,ee); n ∈ nat|] ==> emb(DD`n,DD`succ(n),ee`n)"

(*----------------------------------------------------------------------*)
(* Dinf, the inverse Limit.                                             *)
(*----------------------------------------------------------------------*)

lemma DinfI:
"[|x:(∏n ∈ nat. set(DD`n));
!!n. n ∈ nat ==> Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n|]
==> x ∈ set(Dinf(DD,ee))"
apply (blast intro: mkcpoI iprodI)
done

lemma Dinf_prod: "x ∈ set(Dinf(DD,ee)) ==> x:(∏n ∈ nat. set(DD`n))"
apply (erule mkcpoD1 [THEN iprodE])
done

lemma Dinf_eq:
"[|x ∈ set(Dinf(DD,ee)); n ∈ nat|]
==> Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n"
apply (blast dest: mkcpoD2)
done

lemma rel_DinfI:
"[|!!n. n ∈ nat ==> rel(DD`n,x`n,y`n);
x:(∏n ∈ nat. set(DD`n)); y:(∏n ∈ nat. set(DD`n))|]
==> rel(Dinf(DD,ee),x,y)"
apply (blast intro: rel_mkcpo [THEN iffD2] rel_iprodI iprodI)
done

lemma rel_Dinf: "[|rel(Dinf(DD,ee),x,y); n ∈ nat|] ==> rel(DD`n,x`n,y`n)"
apply (erule rel_mkcpoE [THEN rel_iprodE], assumption)
done

lemma chain_Dinf: "chain(Dinf(DD,ee),X) ==> chain(iprod(DD),X)"
apply (erule chain_mkcpo)
done

lemma subcpo_Dinf: "emb_chain(DD,ee) ==> subcpo(Dinf(DD,ee),iprod(DD))"
apply (rule subcpo_mkcpo)
apply (rule ballI)
apply (simplesubst lub_iprod)
―‹Subst would rewrite the lhs. We want to change the rhs.›
apply (assumption | rule chain_Dinf emb_chain_cpo)+
apply simp
apply (subst Rp_cont [THEN cont_lub])
apply (assumption |
rule emb_chain_cpo emb_chain_emb nat_succI chain_iprod chain_Dinf)+
(* Useful simplification, ugly in HOL. *)
apply (auto intro: cpo_iprod emb_chain_cpo)
done

(* Simple example of existential reasoning in Isabelle versus HOL. *)

lemma cpo_Dinf: "emb_chain(DD,ee) ==> cpo(Dinf(DD,ee))"
apply (rule subcpo_cpo)
apply (erule subcpo_Dinf)
apply (auto intro: cpo_iprod emb_chain_cpo)
done

(* Again and again the proofs are much easier to WRITE in Isabelle, but
the proof steps are essentially the same (I think). *)

lemma lub_Dinf:
"[|chain(Dinf(DD,ee),X); emb_chain(DD,ee)|]
==> lub(Dinf(DD,ee),X) = (λn ∈ nat. lub(DD`n,λm ∈ nat. X`m`n))"
apply (subst subcpo_Dinf [THEN lub_subcpo])
apply (auto intro: cpo_iprod emb_chain_cpo lub_iprod chain_Dinf)
done

(*----------------------------------------------------------------------*)
(* Generalising embedddings D_m -> D_{m+1} to embeddings D_m -> D_n,    *)
(* defined as eps(DD,ee,m,n), via e_less and e_gr.                      *)
(*----------------------------------------------------------------------*)

lemma e_less_eq:
"m ∈ nat ==> e_less(DD,ee,m,m) = id(set(DD`m))"

lemma lemma_succ_sub: "succ(m#+n)#-m = succ(natify(n))"
by simp

"e_less(DD,ee,m,succ(m#+k)) = (ee`(m#+k))O(e_less(DD,ee,m,m#+k))"

lemma le_exists:
"[| m ≤ n;  !!x. [|n=m#+x; x ∈ nat|] ==> Q;  n ∈ nat |] ==> Q"
done

lemma e_less_le: "[| m ≤ n;  n ∈ nat |]
==> e_less(DD,ee,m,succ(n)) = ee`n O e_less(DD,ee,m,n)"
apply (rule le_exists, assumption)
done

(* All theorems assume variables m and n are natural numbers. *)

lemma e_less_succ:
"m ∈ nat ==> e_less(DD,ee,m,succ(m)) = ee`m O id(set(DD`m))"

lemma e_less_succ_emb:
"[|!!n. n ∈ nat ==> emb(DD`n,DD`succ(n),ee`n); m ∈ nat|]
==> e_less(DD,ee,m,succ(m)) = ee`m"
apply (blast intro: emb_cont cont_fun comp_id)
done

(* Compare this proof with the HOL one, here we do type checking. *)
(* In any case the one below was very easy to write. *)

"[| emb_chain(DD,ee); m ∈ nat |]
==> emb(DD`m, DD`(m#+k), e_less(DD,ee,m,m#+k))"
apply (subgoal_tac "emb (DD`m, DD` (m#+natify (k)),
e_less (DD,ee,m,m#+natify (k))) ")
apply (rule_tac [2] n = "natify (k) " in nat_induct)
apply (assumption | rule emb_id emb_chain_cpo)+
apply (auto intro: emb_comp emb_chain_emb emb_chain_cpo)
done

lemma emb_e_less:
"[| m ≤ n;  emb_chain(DD,ee);  n ∈ nat |]
==> emb(DD`m, DD`n, e_less(DD,ee,m,n))"
apply (frule lt_nat_in_nat)
apply (erule nat_succI)
(* same proof as e_less_le *)
apply (rule le_exists, assumption)
done

lemma comp_mono_eq: "[|f=f'; g=g'|] ==> f O g = f' O g'"
by simp

(* Note the object-level implication for induction on k. This
must be removed later to allow the theorems to be used for simp.
Therefore this theorem is only a lemma. *)

"[| emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> n ≤ k ⟶
e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)"
apply (induct_tac k)
apply (simp add: e_less_eq id_type [THEN id_comp])
apply (rule impI)
apply (erule disjE)
apply (erule impE, assumption)
apply (subst e_less_le)
apply (subst comp_assoc)
apply (assumption | rule comp_mono_eq refl)+
apply (subst id_comp) (* simp cannot unify/inst right, use brr below (?) . *)
apply (assumption |
rule emb_e_less_add [THEN emb_cont, THEN cont_fun] refl nat_succI)+
done

"[| n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)"

lemma e_gr_eq:
"m ∈ nat ==> e_gr(DD,ee,m,m) = id(set(DD`m))"

"[|n ∈ nat; k ∈ nat|]
==> e_gr(DD,ee,succ(n#+k),n) =
e_gr(DD,ee,n#+k,n) O Rp(DD`(n#+k),DD`succ(n#+k),ee`(n#+k))"

lemma e_gr_le:
"[|n ≤ m; m ∈ nat; n ∈ nat|]
==> e_gr(DD,ee,succ(m),n) = e_gr(DD,ee,m,n) O Rp(DD`m,DD`succ(m),ee`m)"
apply (erule le_exists)
done

lemma e_gr_succ:
"m ∈ nat ==> e_gr(DD,ee,succ(m),m) = id(set(DD`m)) O Rp(DD`m,DD`succ(m),ee`m)"

(* Cpo asm's due to THE uniqueness. *)
lemma e_gr_succ_emb: "[|emb_chain(DD,ee); m ∈ nat|]
==> e_gr(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)"
apply (blast intro: id_comp Rp_cont cont_fun emb_chain_cpo emb_chain_emb)
done

"[|emb_chain(DD,ee); n ∈ nat; k ∈ nat|]
==> e_gr(DD,ee,n#+k,n): set(DD`(n#+k))->set(DD`n)"
apply (induct_tac k)
apply (blast intro: comp_fun Rp_cont cont_fun emb_chain_emb emb_chain_cpo)
done

lemma e_gr_fun:
"[|n ≤ m; emb_chain(DD,ee); m ∈ nat; n ∈ nat|]
==> e_gr(DD,ee,m,n): set(DD`m)->set(DD`n)"
apply (rule le_exists, assumption)
done

"[| emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> m ≤ k ⟶
e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)"
apply (induct_tac k)
apply (rule impI)
apply (simp add: le0_iff e_gr_eq id_type [THEN comp_id])
apply (rule impI)
apply (erule disjE)
apply (erule impE, assumption)
apply (subst e_gr_le)
apply (subst comp_assoc)
apply (assumption | rule comp_mono_eq refl)+
(* New direct subgoal *)
apply (subst comp_id) (* simp cannot unify/inst right, use brr below (?) . *)
done

"[| m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)"
apply (blast intro: e_gr_split_add_lemma [THEN mp])
done

lemma e_less_cont:
"[|m ≤ n; emb_chain(DD,ee); m ∈ nat; n ∈ nat|]
==> e_less(DD,ee,m,n):cont(DD`m,DD`n)"
apply (blast intro: emb_cont emb_e_less)
done

lemma e_gr_cont:
"[|n ≤ m; emb_chain(DD,ee); m ∈ nat; n ∈ nat|]
==> e_gr(DD,ee,m,n):cont(DD`m,DD`n)"
apply (erule rev_mp)
apply (induct_tac m)
apply (simp add: le0_iff e_gr_eq nat_0I)
apply (assumption | rule impI id_cont emb_chain_cpo nat_0I)+
apply (erule disjE)
apply (erule impE, assumption)
apply (blast intro: comp_pres_cont Rp_cont emb_chain_cpo emb_chain_emb)
done

(* Considerably shorter.... 57 against 26 *)

"[|n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> e_less(DD,ee,m,m#+n) = e_gr(DD,ee,m#+k,m#+n) O e_less(DD,ee,m,m#+k)"
(* Use mp to prepare for induction. *)
apply (erule rev_mp)
apply (induct_tac k)
apply (simp add: e_gr_eq e_less_eq id_type [THEN id_comp])
apply (rule impI)
apply (erule disjE)
apply (erule impE, assumption)
apply (subst comp_assoc)
apply (rule_tac s1 = "ee` (m#+x)" in comp_assoc [THEN subst])
apply (subst embRp_eq)
apply (assumption | rule emb_chain_emb add_type emb_chain_cpo nat_succI)+
apply (subst id_comp)
apply (blast intro: e_less_cont [THEN cont_fun] add_le_self)
apply (rule refl)
apply (blast intro: id_comp [symmetric] e_less_cont [THEN cont_fun]
done

(* Again considerably shorter, and easy to obtain from the previous thm. *)

"[|m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> e_gr(DD,ee,n#+m,n) = e_gr(DD,ee,n#+k,n) O e_less(DD,ee,n#+m,n#+k)"
(* Use mp to prepare for induction. *)
apply (erule rev_mp)
apply (induct_tac k)
apply (simp add: e_gr_eq e_less_eq id_type [THEN id_comp])
apply (rule impI)
apply (erule disjE)
apply (erule impE, assumption)
apply (subst comp_assoc)
apply (rule_tac s1 = "ee` (n#+x)" in comp_assoc [THEN subst])
apply (subst embRp_eq)
apply (assumption | rule emb_chain_emb add_type emb_chain_cpo nat_succI)+
apply (subst id_comp)
apply (blast intro!: e_less_cont [THEN cont_fun] add_le_mono nat_le_refl)
apply (rule refl)
apply (blast intro: comp_id [symmetric] e_gr_cont [THEN cont_fun] add_le_self)
done

lemma emb_eps:
"[|m ≤ n; emb_chain(DD,ee); m ∈ nat; n ∈ nat|]
==> emb(DD`m,DD`n,eps(DD,ee,m,n))"
apply (blast intro: emb_e_less)
done

lemma eps_fun:
"[|emb_chain(DD,ee); m ∈ nat; n ∈ nat|]
==> eps(DD,ee,m,n): set(DD`m)->set(DD`n)"
apply (auto intro: e_less_cont [THEN cont_fun]
not_le_iff_lt [THEN iffD1, THEN leI]
e_gr_fun nat_into_Ord)
done

lemma eps_id: "n ∈ nat ==> eps(DD,ee,n,n) = id(set(DD`n))"

"[|m ∈ nat; n ∈ nat|] ==> eps(DD,ee,m,m#+n) = e_less(DD,ee,m,m#+n)"

lemma eps_e_less:
"[|m ≤ n; m ∈ nat; n ∈ nat|] ==> eps(DD,ee,m,n) = e_less(DD,ee,m,n)"

"[|n ∈ nat; k ∈ nat|] ==> eps(DD,ee,n#+k,n) = e_gr(DD,ee,n#+k,n)"
by (simp add: eps_def e_less_eq e_gr_eq)

lemma eps_e_gr:
"[|n ≤ m; m ∈ nat; n ∈ nat|] ==> eps(DD,ee,m,n) = e_gr(DD,ee,m,n)"
apply (erule le_exists)
done

lemma eps_succ_ee:
"[|!!n. n ∈ nat ==> emb(DD`n,DD`succ(n),ee`n); m ∈ nat|]
==> eps(DD,ee,m,succ(m)) = ee`m"
by (simp add: eps_e_less le_succ_iff e_less_succ_emb)

lemma eps_succ_Rp:
"[|emb_chain(DD,ee); m ∈ nat|]
==> eps(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)"
by (simp add: eps_e_gr le_succ_iff e_gr_succ_emb)

lemma eps_cont:
"[|emb_chain(DD,ee); m ∈ nat; n ∈ nat|] ==> eps(DD,ee,m,n): cont(DD`m,DD`n)"
apply (rule_tac i = m and j = n in nat_linear_le)
apply (simp_all add: eps_e_less e_less_cont eps_e_gr e_gr_cont)
done

"[|n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,m#+k) = eps(DD,ee,m#+n,m#+k) O eps(DD,ee,m,m#+n)"
done

"[|n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,m#+n) = eps(DD,ee,m#+k,m#+n) O eps(DD,ee,m,m#+k)"
done

"[|m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,n#+k,n) = eps(DD,ee,n#+m,n) O eps(DD,ee,n#+k,n#+m)"
done

"[|m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,n#+m,n) = eps(DD,ee,n#+k,n) O eps(DD,ee,n#+m,n#+k)"
done

(* Arithmetic *)

lemma le_exists_lemma:
"[| n ≤ k; k ≤ m;
!!p q. [|p ≤ q; k=n#+p; m=n#+q; p ∈ nat; q ∈ nat|] ==> R;
m ∈ nat |]==>R"
apply (rule le_exists, assumption)
prefer 2 apply (simp add: lt_nat_in_nat)
apply (rule le_trans [THEN le_exists], assumption+, force+)
done

lemma eps_split_left_le:
"[|m ≤ k; k ≤ n; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule le_exists_lemma, assumption+)
done

lemma eps_split_left_le_rev:
"[|m ≤ n; n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule le_exists_lemma, assumption+)
done

lemma eps_split_right_le:
"[|n ≤ k; k ≤ m; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule le_exists_lemma, assumption+)
done

lemma eps_split_right_le_rev:
"[|n ≤ m; m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule le_exists_lemma, assumption+)
done

(* The desired two theorems about `splitting'. *)

lemma eps_split_left:
"[|m ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule nat_linear_le)
apply (rule_tac [4] eps_split_right_le_rev)
prefer 4 apply assumption
apply (rule_tac [3] nat_linear_le)
apply (rule_tac [5] eps_split_left_le)
prefer 6 apply assumption
done

lemma eps_split_right:
"[|n ≤ k; emb_chain(DD,ee); m ∈ nat; n ∈ nat; k ∈ nat|]
==> eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)"
apply (rule nat_linear_le)
apply (rule_tac [3] eps_split_left_le_rev)
prefer 3 apply assumption
apply (rule_tac [8] nat_linear_le)
apply (rule_tac [10] eps_split_right_le)
prefer 11 apply assumption
done

(*----------------------------------------------------------------------*)
(* That was eps: D_m -> D_n, NEXT rho_emb: D_n -> Dinf.                 *)
(*----------------------------------------------------------------------*)

(* Considerably shorter. *)

lemma rho_emb_fun:
"[|emb_chain(DD,ee); n ∈ nat|]
==> rho_emb(DD,ee,n): set(DD`n) -> set(Dinf(DD,ee))"
apply (assumption |
rule lam_type DinfI eps_cont [THEN cont_fun, THEN apply_type])+
apply simp
apply (rule_tac i = "succ (na) " and j = n in nat_linear_le)
apply blast
apply assumption
apply (simplesubst eps_split_right_le)
―‹Subst would rewrite the lhs. We want to change the rhs.›
prefer 2 apply assumption
apply simp
apply (assumption | rule add_le_self nat_0I nat_succI)+
apply (subst comp_fun_apply)
apply (assumption |
rule eps_fun nat_succI Rp_cont [THEN cont_fun]
emb_chain_emb emb_chain_cpo refl)+
(* Now the second part of the proof. Slightly different than HOL. *)
apply (erule le_iff [THEN iffD1, THEN disjE])
apply (subst comp_fun_apply)
apply (assumption | rule e_less_cont cont_fun emb_chain_emb emb_cont)+
apply (subst embRp_eq_thm)
apply (assumption |
rule emb_chain_emb e_less_cont [THEN cont_fun, THEN apply_type]
emb_chain_cpo nat_succI)+
apply (simp add: eps_succ_Rp e_less_eq id_conv nat_succI)
done

lemma rho_emb_apply1:
"x ∈ set(DD`n) ==> rho_emb(DD,ee,n)`x = (λm ∈ nat. eps(DD,ee,n,m)`x)"

lemma rho_emb_apply2:
"[|x ∈ set(DD`n); m ∈ nat|] ==> rho_emb(DD,ee,n)`x`m = eps(DD,ee,n,m)`x"

lemma rho_emb_id: "[| x ∈ set(DD`n); n ∈ nat|] ==> rho_emb(DD,ee,n)`x`n = x"

(* Shorter proof, 23 against 62. *)

lemma rho_emb_cont:
"[|emb_chain(DD,ee); n ∈ nat|]
==> rho_emb(DD,ee,n): cont(DD`n,Dinf(DD,ee))"
apply (rule contI)
apply (assumption | rule rho_emb_fun)+
apply (rule rel_DinfI)
apply (assumption |
rule eps_cont [THEN cont_mono]  Dinf_prod apply_type rho_emb_fun)+
(* Continuity, different order, slightly different proofs. *)
apply (subst lub_Dinf)
apply (rule chainI)
apply (assumption | rule lam_type rho_emb_fun [THEN apply_type]  chain_in)+
apply simp
apply (rule rel_DinfI)
apply (assumption |
rule eps_cont [THEN cont_mono]  chain_rel Dinf_prod
rho_emb_fun [THEN apply_type]  chain_in nat_succI)+
(* Now, back to the result of applying lub_Dinf *)
apply (subst rho_emb_apply1)
apply (assumption | rule cpo_lub [THEN islub_in]  emb_chain_cpo)+
apply (rule fun_extension)
apply (assumption |
rule lam_type eps_cont [THEN cont_fun, THEN apply_type]
cpo_lub [THEN islub_in]  emb_chain_cpo)+
apply (assumption | rule cont_chain eps_cont emb_chain_cpo)+
apply simp
apply (simp add: eps_cont [THEN cont_lub])
done

(* 32 vs 61, using safe_tac with imp in asm would be unfortunate (5steps) *)

lemma eps1_aux1:
"[|m ≤ n; emb_chain(DD,ee); x ∈ set(Dinf(DD,ee)); m ∈ nat; n ∈ nat|]
==> rel(DD`n,e_less(DD,ee,m,n)`(x`m),x`n)"
apply (erule rev_mp) (* For induction proof *)
apply (induct_tac n)
apply (rule impI)
apply (subst id_conv)
apply (assumption | rule apply_type Dinf_prod cpo_refl emb_chain_cpo nat_0I)+
apply (rule impI)
apply (erule disjE)
apply (drule mp, assumption)
apply (rule cpo_trans)
apply (rule_tac [2] e_less_le [THEN ssubst])
apply (assumption | rule emb_chain_cpo nat_succI)+
apply (subst comp_fun_apply)
apply (assumption |
rule emb_chain_emb [THEN emb_cont]  e_less_cont cont_fun apply_type
Dinf_prod)+
apply (rule_tac y = "x`xa" in emb_chain_emb [THEN emb_cont, THEN cont_mono])
apply (assumption | rule e_less_cont [THEN cont_fun]  apply_type Dinf_prod)+
apply (rule_tac x1 = x and n1 = xa in Dinf_eq [THEN subst])
apply (rule_tac [3] comp_fun_apply [THEN subst])
apply (rename_tac [5] y)
apply (rule_tac [5] P =
"%z. rel(DD`succ(y),
(ee`y O Rp(DD'(y)`y,DD'(y)`succ(y),ee'(y)`y)) ` (x`succ(y)),
z)" for DD' ee'
in id_conv [THEN subst])
apply (rule_tac [6] rel_cf)
(* Dinf and cont_fun doesn't go well together, both Pi(_,%x._). *)
(* solves 10 of 11 subgoals *)
apply (assumption |
rule Dinf_prod [THEN apply_type] cont_fun Rp_cont e_less_cont
emb_cont emb_chain_emb emb_chain_cpo apply_type embRp_rel
disjI1 [THEN le_succ_iff [THEN iffD2]]  nat_succI)+
apply (subst id_conv)
apply (auto intro: apply_type Dinf_prod emb_chain_cpo)
done

(* 18 vs 40 *)

lemma eps1_aux2:
"[|n ≤ m; emb_chain(DD,ee); x ∈ set(Dinf(DD,ee)); m ∈ nat; n ∈ nat|]
==> rel(DD`n,e_gr(DD,ee,m,n)`(x`m),x`n)"
apply (erule rev_mp) (* For induction proof *)
apply (induct_tac m)
apply (rule impI)
apply (subst id_conv)
apply (assumption | rule apply_type Dinf_prod cpo_refl emb_chain_cpo nat_0I)+
apply (rule impI)
apply (erule disjE)
apply (drule mp, assumption)
apply (subst e_gr_le)
apply (rule_tac [4] comp_fun_apply [THEN ssubst])
apply (rule_tac [6] Dinf_eq [THEN ssubst])
apply (assumption |
rule emb_chain_emb emb_chain_cpo Rp_cont e_gr_cont cont_fun emb_cont
apply_type Dinf_prod nat_succI)+
apply (subst id_conv)
apply (auto intro: apply_type Dinf_prod emb_chain_cpo)
done

lemma eps1:
"[|emb_chain(DD,ee); x ∈ set(Dinf(DD,ee)); m ∈ nat; n ∈ nat|]
==> rel(DD`n,eps(DD,ee,m,n)`(x`m),x`n)"
apply (blast intro: eps1_aux1 not_le_iff_lt [THEN iffD1, THEN leI, THEN eps1_aux2]
nat_into_Ord)
done

(* The following theorem is needed/useful due to type check for rel_cfI,
but also elsewhere.
Look for occurrences of rel_cfI, rel_DinfI, etc to evaluate the problem. *)

lemma lam_Dinf_cont:
"[| emb_chain(DD,ee); n ∈ nat |]
==> (λx ∈ set(Dinf(DD,ee)). x`n) ∈ cont(Dinf(DD,ee),DD`n)"
apply (rule contI)
apply (assumption | rule lam_type apply_type Dinf_prod)+
apply simp
apply (assumption | rule rel_Dinf)+
apply (subst beta)
apply (auto intro: cpo_Dinf islub_in cpo_lub)
done

lemma rho_projpair:
"[| emb_chain(DD,ee); n ∈ nat |]
==> projpair(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n),rho_proj(DD,ee,n))"
apply (rule projpairI)
apply (assumption | rule rho_emb_cont)+
(* lemma used, introduced because same fact needed below due to rel_cfI. *)
apply (assumption | rule lam_Dinf_cont)+
(*-----------------------------------------------*)
(* This part is 7 lines, but 30 in HOL (75% reduction!) *)
apply (rule fun_extension)
apply (rule_tac [3] id_conv [THEN ssubst])
apply (rule_tac [4] comp_fun_apply [THEN ssubst])
apply (rule_tac [6] beta [THEN ssubst])
apply (rule_tac [7] rho_emb_id [THEN ssubst])
apply (assumption |
rule comp_fun id_type lam_type rho_emb_fun Dinf_prod [THEN apply_type]
apply_type refl)+
(*^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
apply (rule rel_cfI) (* ----------------⟶>>Yields type cond, not in HOL *)
apply (subst id_conv)
apply (rule_tac [2] comp_fun_apply [THEN ssubst])
apply (rule_tac [4] beta [THEN ssubst])
apply (rule_tac [5] rho_emb_apply1 [THEN ssubst])
apply (rule_tac [6] rel_DinfI)
apply (rule_tac [6] beta [THEN ssubst])
(* Dinf_prod bad with lam_type *)
apply (assumption |
rule eps1 lam_type rho_emb_fun eps_fun
Dinf_prod [THEN apply_type] refl)+
apply (assumption |
rule apply_type eps_fun Dinf_prod comp_pres_cont rho_emb_cont
lam_Dinf_cont id_cont cpo_Dinf emb_chain_cpo)+
done

lemma emb_rho_emb:
"[| emb_chain(DD,ee); n ∈ nat |] ==> emb(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))"
by (auto simp add: emb_def intro: exI rho_projpair)

lemma rho_proj_cont: "[| emb_chain(DD,ee); n ∈ nat |]
==> rho_proj(DD,ee,n) ∈ cont(Dinf(DD,ee),DD`n)"
by (auto intro: rho_projpair projpair_p_cont)

(*----------------------------------------------------------------------*)
(* Commutivity and universality.                                        *)
(*----------------------------------------------------------------------*)

lemma commuteI:
"[| !!n. n ∈ nat ==> emb(DD`n,E,r(n));
!!m n. [|m ≤ n; m ∈ nat; n ∈ nat|] ==> r(n) O eps(DD,ee,m,n) = r(m) |]
==> commute(DD,ee,E,r)"

lemma commute_emb [TC]:
"[| commute(DD,ee,E,r); n ∈ nat |] ==> emb(DD`n,E,r(n))"

lemma commute_eq:
"[| commute(DD,ee,E,r); m ≤ n; m ∈ nat; n ∈ nat |]
==> r(n) O eps(DD,ee,m,n) = r(m) "

(* Shorter proof: 11 vs 46 lines. *)

lemma rho_emb_commute:
"emb_chain(DD,ee) ==> commute(DD,ee,Dinf(DD,ee),rho_emb(DD,ee))"
apply (rule commuteI)
apply (assumption | rule emb_rho_emb)+
apply (rule fun_extension) (* Manual instantiation in HOL. *)
apply (rule_tac [3] comp_fun_apply [THEN ssubst])
apply (rule_tac [5] fun_extension) (*Next, clean up and instantiate unknowns *)
apply (assumption | rule comp_fun rho_emb_fun eps_fun Dinf_prod apply_type)+
apply (simp add: rho_emb_apply2 eps_fun [THEN apply_type])
apply (rule comp_fun_apply [THEN subst])
apply (rule_tac [3] eps_split_left [THEN subst])
apply (auto intro: eps_fun)
done

lemma le_succ: "n ∈ nat ==> n ≤ succ(n)"

(* Shorter proof: 21 vs 83 (106 - 23, due to OAssoc complication) *)

lemma commute_chain:
"[| commute(DD,ee,E,r); emb_chain(DD,ee); cpo(E) |]
==> chain(cf(E,E),λn ∈ nat. r(n) O Rp(DD`n,E,r(n)))"
apply (rule chainI)
apply (blast intro: lam_type cont_cf comp_pres_cont commute_emb Rp_cont
emb_cont emb_chain_cpo,
simp)
apply (rule_tac r1 = r and m1 = n in commute_eq [THEN subst])
apply (assumption | rule le_succ nat_succI)+
apply (subst Rp_comp)
apply (assumption | rule emb_eps commute_emb emb_chain_cpo le_succ nat_succI)+
apply (rule comp_assoc [THEN subst]) (* comp_assoc is simpler in Isa *)
apply (rule_tac r1 = "r (succ (n))" in comp_assoc [THEN ssubst])
apply (rule comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps commute_emb Rp_cont
emb_cont emb_chain_cpo le_succ)+
apply (rule_tac b="r(succ(n))" in comp_id [THEN subst]) (* 1 subst too much *)
apply (rule_tac [2] comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps emb_id commute_emb
Rp_cont emb_cont cont_fun emb_chain_cpo le_succ)+
apply (subst comp_id) (* Undoes "1 subst too much", typing next anyway *)
apply (blast intro: cont_fun Rp_cont emb_cont commute_emb cont_cf cpo_cf
emb_chain_cpo embRp_rel emb_eps le_succ)+
done

lemma rho_emb_chain:
"emb_chain(DD,ee) ==>
chain(cf(Dinf(DD,ee),Dinf(DD,ee)),
λn ∈ nat. rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))"
by (auto intro: commute_chain rho_emb_commute cpo_Dinf)

lemma rho_emb_chain_apply1:
"[| emb_chain(DD,ee); x ∈ set(Dinf(DD,ee)) |]
==> chain(Dinf(DD,ee),
λn ∈ nat.
(rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))`x)"
by (drule rho_emb_chain [THEN chain_cf], assumption, simp)

lemma chain_iprod_emb_chain:
"[| chain(iprod(DD),X); emb_chain(DD,ee); n ∈ nat |]
==> chain(DD`n,λm ∈ nat. X `m `n)"
by (auto intro: chain_iprod emb_chain_cpo)

lemma rho_emb_chain_apply2:
"[| emb_chain(DD,ee); x ∈ set(Dinf(DD,ee)); n ∈ nat |] ==>
chain
(DD`n,
λxa ∈ nat.
(rho_emb(DD, ee, xa) O Rp(DD ` xa, Dinf(DD, ee),rho_emb(DD, ee, xa))) `
x ` n)"
by (frule rho_emb_chain_apply1 [THEN chain_Dinf, THEN chain_iprod_emb_chain],
auto)

(* Shorter proof: 32 vs 72 (roughly), Isabelle proof has lemmas. *)

lemma rho_emb_lub:
"emb_chain(DD,ee) ==>
lub(cf(Dinf(DD,ee),Dinf(DD,ee)),
λn ∈ nat. rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))) =
id(set(Dinf(DD,ee)))"
apply (rule cpo_antisym)
apply (rule cpo_cf) (*Instantiate variable, continued below (loops otherwise)*)
apply (assumption | rule cpo_Dinf)+
apply (rule islub_least)
apply (assumption |
rule cpo_lub rho_emb_chain cpo_cf cpo_Dinf isubI cont_cf id_cont)+
apply simp
apply (assumption | rule embRp_rel emb_rho_emb emb_chain_cpo cpo_Dinf)+
apply (rule rel_cfI)
apply (simp add: lub_cf rho_emb_chain cpo_Dinf)
apply (rule rel_DinfI) (* Additional assumptions *)
apply (subst lub_Dinf)
apply (assumption | rule rho_emb_chain_apply1)+
defer 1
apply (assumption |
rule Dinf_prod cpo_lub [THEN islub_in]  id_cont cpo_Dinf cpo_cf cf_cont
rho_emb_chain rho_emb_chain_apply1 id_cont [THEN cont_cf])+
apply simp
apply (rule dominate_islub)
apply (rule_tac [3] cpo_lub)
apply (rule_tac [6] x1 = "x`n" in chain_const [THEN chain_fun])
defer 1
apply (assumption |
rule rho_emb_chain_apply2 emb_chain_cpo islub_const apply_type
Dinf_prod emb_chain_cpo chain_fun rho_emb_chain_apply2)+
apply (rule dominateI, assumption, simp)
apply (subst comp_fun_apply)
apply (assumption |
rule cont_fun Rp_cont emb_cont emb_rho_emb cpo_Dinf emb_chain_cpo)+
apply (subst rho_projpair [THEN Rp_unique])
prefer 5
apply (rule rho_emb_id [THEN ssubst])
apply (auto intro: cpo_Dinf apply_type Dinf_prod emb_chain_cpo)
done

lemma theta_chain: (* almost same proof as commute_chain *)
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> chain(cf(E,G),λn ∈ nat. f(n) O Rp(DD`n,E,r(n)))"
apply (rule chainI)
apply (blast intro: lam_type cont_cf comp_pres_cont commute_emb Rp_cont
emb_cont emb_chain_cpo,
simp)
apply (rule_tac r1 = r and m1 = n in commute_eq [THEN subst])
apply (rule_tac [5] r1 = f and m1 = n in commute_eq [THEN subst])
apply (assumption | rule le_succ nat_succI)+
apply (subst Rp_comp)
apply (assumption | rule emb_eps commute_emb emb_chain_cpo le_succ nat_succI)+
apply (rule comp_assoc [THEN subst])
apply (rule_tac r1 = "f (succ (n))" in comp_assoc [THEN ssubst])
apply (rule comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps commute_emb Rp_cont
emb_cont emb_chain_cpo le_succ)+
apply (rule_tac b="f(succ(n))" in comp_id [THEN subst]) (* 1 subst too much *)
apply (rule_tac [2] comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps emb_id commute_emb
Rp_cont emb_cont cont_fun emb_chain_cpo le_succ)+
apply (subst comp_id) (* Undoes "1 subst too much", typing next anyway *)
apply (blast intro: cont_fun Rp_cont emb_cont commute_emb cont_cf cpo_cf
emb_chain_cpo embRp_rel emb_eps le_succ)+
done

lemma theta_proj_chain: (* similar proof to theta_chain *)
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> chain(cf(G,E),λn ∈ nat. r(n) O Rp(DD`n,G,f(n)))"
apply (rule chainI)
apply (blast intro: lam_type cont_cf comp_pres_cont commute_emb Rp_cont
emb_cont emb_chain_cpo, simp)
apply (rule_tac r1 = r and m1 = n in commute_eq [THEN subst])
apply (rule_tac [5] r1 = f and m1 = n in commute_eq [THEN subst])
apply (assumption | rule le_succ nat_succI)+
apply (subst Rp_comp)
apply (assumption | rule emb_eps commute_emb emb_chain_cpo le_succ nat_succI)+
apply (rule comp_assoc [THEN subst]) (* comp_assoc is simpler in Isa *)
apply (rule_tac r1 = "r (succ (n))" in comp_assoc [THEN ssubst])
apply (rule comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps commute_emb Rp_cont
emb_cont emb_chain_cpo le_succ)+
apply (rule_tac b="r(succ(n))" in comp_id [THEN subst]) (* 1 subst too much *)
apply (rule_tac [2] comp_mono)
apply (blast intro: comp_pres_cont eps_cont emb_eps emb_id commute_emb
Rp_cont emb_cont cont_fun emb_chain_cpo le_succ)+
apply (subst comp_id) (* Undoes "1 subst too much", typing next anyway *)
apply (blast intro: cont_fun Rp_cont emb_cont commute_emb cont_cf cpo_cf
emb_chain_cpo embRp_rel emb_eps le_succ)+
done

(* Simplification with comp_assoc is possible inside a λ-abstraction,
because it does not have assumptions. If it had, as the HOL-ST theorem
too strongly has, we would be in deep trouble due to HOL's lack of proper
conditional rewriting (a HOL contrib provides something that works). *)

(* Controlled simplification inside lambda: introduce lemmas *)

lemma commute_O_lemma:
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G); x ∈ nat |]
==> r(x) O Rp(DD ` x, G, f(x)) O f(x) O Rp(DD ` x, E, r(x)) =
r(x) O Rp(DD ` x, E, r(x))"
apply (rule_tac s1 = "f (x) " in comp_assoc [THEN subst])
apply (subst embRp_eq)
apply (rule_tac [4] id_comp [THEN ssubst])
apply (auto intro: cont_fun Rp_cont commute_emb emb_chain_cpo)
done

(* Shorter proof (but lemmas): 19 vs 79 (103 - 24, due to OAssoc)  *)

lemma theta_projpair:
"[| lub(cf(E,E), λn ∈ nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E));
commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> projpair
(E,G,
lub(cf(E,G), λn ∈ nat. f(n) O Rp(DD`n,E,r(n))),
lub(cf(G,E), λn ∈ nat. r(n) O Rp(DD`n,G,f(n))))"
apply (simp add: projpair_def rho_proj_def, safe)
apply (rule_tac [3] comp_lubs [THEN ssubst])
(* The following one line is 15 lines in HOL, and includes existentials. *)
apply (assumption |
rule cf_cont islub_in cpo_lub cpo_cf theta_chain theta_proj_chain)+
apply (subst comp_lubs)
apply (assumption |
rule cf_cont islub_in cpo_lub cpo_cf theta_chain theta_proj_chain)+
apply (rule dominate_islub)
defer 1
apply (rule cpo_lub)
apply (assumption |
rule commute_chain commute_emb islub_const cont_cf id_cont
cpo_cf chain_fun chain_const)+
apply (rule dominateI, assumption, simp)
apply (blast intro: embRp_rel commute_emb emb_chain_cpo)
done

lemma emb_theta:
"[| lub(cf(E,E), λn ∈ nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E));
commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> emb(E,G,lub(cf(E,G), λn ∈ nat. f(n) O Rp(DD`n,E,r(n))))"
apply (blast intro: theta_projpair)
done

lemma mono_lemma:
"[| g ∈ cont(D,D'); cpo(D); cpo(D'); cpo(E) |]
==> (λf ∈ cont(D',E). f O g) ∈ mono(cf(D',E),cf(D,E))"
apply (rule monoI)
apply (drule cf_cont)+
apply simp
apply (blast intro: comp_mono lam_type comp_pres_cont cpo_cf cont_cf)
done

lemma commute_lam_lemma:
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G); n ∈ nat |]
==> (λna ∈ nat. (λf ∈ cont(E, G). f O r(n)) `
((λn ∈ nat. f(n) O Rp(DD ` n, E, r(n))) ` na))  =
(λna ∈ nat. (f(na) O Rp(DD ` na, E, r(na))) O r(n))"
apply (rule fun_extension)
(*something wrong here*)
apply (auto simp del: beta_if simp add: beta intro: lam_type)
done

lemma chain_lemma:
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G); n ∈ nat |]
==> chain(cf(DD`n,G),λx ∈ nat. (f(x) O Rp(DD ` x, E, r(x))) O r(n))"
apply (rule commute_lam_lemma [THEN subst])
apply (blast intro: theta_chain emb_chain_cpo
commute_emb [THEN emb_cont, THEN mono_lemma, THEN mono_chain])+
done

lemma suffix_lemma:
"[| commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G); cpo(DD`x); x ∈ nat |]
==> suffix(λn ∈ nat. (f(n) O Rp(DD`n,E,r(n))) O r(x),x) =
(λn ∈ nat. f(x))"
apply (rule lam_type [THEN fun_extension])
apply (blast intro: lam_type comp_fun cont_fun Rp_cont emb_cont
commute_emb emb_chain_cpo)+
apply simp
apply (rename_tac y)
apply (subgoal_tac
"f(x#+y) O (Rp(DD`(x#+y), E, r(x#+y)) O r (x#+y)) O eps(DD, ee, x, x#+y)
= f(x)")
apply (simp add: embRp_eq eps_fun [THEN id_comp] commute_emb emb_chain_cpo)
done

lemma mediatingI:
"[|emb(E,G,t);  !!n. n ∈ nat ==> f(n) = t O r(n) |]==>mediating(E,G,r,f,t)"

lemma mediating_emb: "mediating(E,G,r,f,t) ==> emb(E,G,t)"

lemma mediating_eq: "[| mediating(E,G,r,f,t); n ∈ nat |] ==> f(n) = t O r(n)"

lemma lub_universal_mediating:
"[| lub(cf(E,E), λn ∈ nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E));
commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> mediating(E,G,r,f,lub(cf(E,G), λn ∈ nat. f(n) O Rp(DD`n,E,r(n))))"
apply (assumption | rule mediatingI emb_theta)+
apply (rule_tac b = "r (n) " in lub_const [THEN subst])
apply (rule_tac [3] comp_lubs [THEN ssubst])
apply (blast intro: cont_cf emb_cont commute_emb cpo_cf theta_chain
chain_const emb_chain_cpo)+
apply (simp (no_asm))
apply (rule_tac n1 = n in lub_suffix [THEN subst])
apply (assumption | rule chain_lemma cpo_cf emb_chain_cpo)+
apply (simp add: suffix_lemma lub_const cont_cf emb_cont commute_emb cpo_cf
emb_chain_cpo)
done

lemma lub_universal_unique:
"[| mediating(E,G,r,f,t);
lub(cf(E,E), λn ∈ nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E));
commute(DD,ee,E,r); commute(DD,ee,G,f);
emb_chain(DD,ee); cpo(E); cpo(G) |]
==> t = lub(cf(E,G), λn ∈ nat. f(n) O Rp(DD`n,E,r(n)))"
apply (rule_tac b = t in comp_id [THEN subst])
apply (erule_tac [2] subst)
apply (rule_tac [2] b = t in lub_const [THEN subst])
apply (rule_tac [4] comp_lubs [THEN ssubst])
prefer 9 apply (simp add: comp_assoc mediating_eq)
apply (assumption |
rule cont_fun emb_cont mediating_emb cont_cf cpo_cf chain_const
commute_chain emb_chain_cpo)+
done

(*---------------------------------------------------------------------*)
(* Dinf yields the inverse_limit, stated as rho_emb_commute and        *)
(* Dinf_universal.                                                     *)
(*---------------------------------------------------------------------*)

theorem Dinf_universal:
"[| commute(DD,ee,G,f); emb_chain(DD,ee); cpo(G) |] ==>
mediating
(Dinf(DD,ee),G,rho_emb(DD,ee),f,
lub(cf(Dinf(DD,ee),G),
λn ∈ nat. f(n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))) &
(∀t. mediating(Dinf(DD,ee),G,rho_emb(DD,ee),f,t) ⟶
t = lub(cf(Dinf(DD,ee),G),
λn ∈ nat. f(n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))))"
apply safe
apply (assumption |
rule lub_universal_mediating rho_emb_commute rho_emb_lub cpo_Dinf)+
apply (auto intro: lub_universal_unique rho_emb_commute rho_emb_lub cpo_Dinf)
done

end
```