# Theory Map

theory Map
imports ZF
```(*  Title:      ZF/Coind/Map.thy
Author:     Jacob Frost, Cambridge University Computer Laboratory

Some sample proofs of inclusions for the final coalgebra "U" (by lcp).
*)

theory Map imports ZF begin

definition
TMap :: "[i,i] => i"  where
"TMap(A,B) == {m ∈ Pow(A*⋃(B)).∀a ∈ A. m``{a} ∈ B}"

definition
PMap :: "[i,i] => i"  where
"PMap(A,B) == TMap(A,cons(0,B))"

(* Note: 0 ∈ B ==> TMap(A,B) = PMap(A,B) *)

definition
map_emp :: i  where
"map_emp == 0"

definition
map_owr :: "[i,i,i]=>i"  where
"map_owr(m,a,b) == ∑x ∈ {a} ∪ domain(m). if x=a then b else m``{x}"

definition
map_app :: "[i,i]=>i"  where
"map_app(m,a) == m``{a}"

lemma "{0,1} ⊆ {1} ∪ TMap(I, {0,1})"
by (unfold TMap_def, blast)

lemma "{0} ∪ TMap(I,1) ⊆ {1} ∪ TMap(I, {0} ∪ TMap(I,1))"
by (unfold TMap_def, blast)

lemma "{0,1} ∪ TMap(I,2) ⊆ {1} ∪ TMap(I, {0,1} ∪ TMap(I,2))"
by (unfold TMap_def, blast)

(*A bit too slow.
lemma "{0,1} ∪ TMap(I,TMap(I,2)) ∪ TMap(I,2) ⊆
{1} ∪ TMap(I, {0,1} ∪ TMap(I,TMap(I,2)) ∪ TMap(I,2))"
by (unfold TMap_def, blast)
*)

(* ############################################################ *)
(* Lemmas                                                       *)
(* ############################################################ *)

lemma qbeta_if: "Sigma(A,B)``{a} = (if a ∈ A then B(a) else 0)"
by auto

lemma qbeta: "a ∈ A ==> Sigma(A,B)``{a} = B(a)"
by fast

lemma qbeta_emp: "a∉A ==> Sigma(A,B)``{a} = 0"
by fast

lemma image_Sigma1: "a ∉ A ==> Sigma(A,B)``{a}=0"
by fast

(* ############################################################ *)
(* Inclusion in Quine Universes                                 *)
(* ############################################################ *)

(* Lemmas *)

lemma MapQU_lemma:
"A ⊆ univ(X) ==> Pow(A * ⋃(quniv(X))) ⊆ quniv(X)"
apply (unfold quniv_def)
apply (rule Pow_mono)
apply (rule subset_trans [OF Sigma_mono product_univ])
apply (erule subset_trans)
apply (rule arg_subset_eclose [THEN univ_mono])
done

(* Theorems *)

lemma mapQU:
"[| m ∈ PMap(A,quniv(B)); !!x. x ∈ A ==> x ∈ univ(B) |] ==> m ∈ quniv(B)"
apply (unfold PMap_def TMap_def)
apply (blast intro!: MapQU_lemma [THEN subsetD])
done

(* ############################################################ *)
(* Monotonicity                                                 *)
(* ############################################################ *)

lemma PMap_mono: "B ⊆ C ==> PMap(A,B)<=PMap(A,C)"
by (unfold PMap_def TMap_def, blast)

(* ############################################################ *)
(* Introduction Rules                                           *)
(* ############################################################ *)

(** map_emp **)

lemma pmap_empI: "map_emp ∈ PMap(A,B)"
by (unfold map_emp_def PMap_def TMap_def, auto)

(** map_owr **)

lemma pmap_owrI:
"[| m ∈ PMap(A,B); a ∈ A; b ∈ B |]  ==> map_owr(m,a,b):PMap(A,B)"
apply (unfold map_owr_def PMap_def TMap_def, safe)
(*Remaining subgoal*)
apply (rule excluded_middle [THEN disjE])
apply (erule image_Sigma1)
apply (drule_tac psi = "uu ∉ B" for uu in asm_rl)
done

(** map_app **)

lemma tmap_app_notempty:
"[| m ∈ TMap(A,B); a ∈ domain(m) |] ==> map_app(m,a) ≠0"
by (unfold TMap_def map_app_def, blast)

lemma tmap_appI:
"[| m ∈ TMap(A,B); a ∈ domain(m) |] ==> map_app(m,a):B"
by (unfold TMap_def map_app_def domain_def, blast)

lemma pmap_appI:
"[| m ∈ PMap(A,B); a ∈ domain(m) |] ==> map_app(m,a):B"
apply (unfold PMap_def)
apply (frule tmap_app_notempty, assumption)
apply (drule tmap_appI, auto)
done

(** domain **)

lemma tmap_domainD:
"[| m ∈ TMap(A,B); a ∈ domain(m) |] ==> a ∈ A"
by (unfold TMap_def, blast)

lemma pmap_domainD:
"[| m ∈ PMap(A,B); a ∈ domain(m) |] ==> a ∈ A"
by (unfold PMap_def TMap_def, blast)

(* ############################################################ *)
(* Equalities                                                   *)
(* ############################################################ *)

(** Domain **)

(* Lemmas *)

lemma domain_UN: "domain(⋃x ∈ A. B(x)) = (⋃x ∈ A. domain(B(x)))"
by fast

lemma domain_Sigma: "domain(Sigma(A,B)) = {x ∈ A. ∃y. y ∈ B(x)}"
by blast

(* Theorems *)

lemma map_domain_emp: "domain(map_emp) = 0"
by (unfold map_emp_def, blast)

lemma map_domain_owr:
"b ≠ 0 ==> domain(map_owr(f,a,b)) = {a} ∪ domain(f)"
apply (unfold map_owr_def)
done

(** Application **)

lemma map_app_owr:
"map_app(map_owr(f,a,b),c) = (if c=a then b else map_app(f,c))"
by (simp add: qbeta_if  map_app_def map_owr_def, blast)

lemma map_app_owr1: "map_app(map_owr(f,a,b),a) = b"