header{*UNITY Program States*}
theory State imports Main begin
consts var :: i
datatype var = Var("i ∈ list(nat)")
type_intros nat_subset_univ [THEN list_subset_univ, THEN subsetD]
consts
type_of :: "i=>i"
default_val :: "i=>i"
definition
"state == Π x ∈ var. cons(default_val(x), type_of(x))"
definition
"st0 == λx ∈ var. default_val(x)"
definition
st_set :: "i=>o" where
"st_set(A) == A<=state"
definition
st_compl :: "i=>i" where
"st_compl(A) == state-A"
lemma st0_in_state [simp,TC]: "st0 ∈ state"
by (simp add: state_def st0_def)
lemma st_set_Collect [iff]: "st_set({x ∈ state. P(x)})"
by (simp add: st_set_def, auto)
lemma st_set_0 [iff]: "st_set(0)"
by (simp add: st_set_def)
lemma st_set_state [iff]: "st_set(state)"
by (simp add: st_set_def)
lemma st_set_Un_iff [iff]: "st_set(A Un B) <-> st_set(A) & st_set(B)"
by (simp add: st_set_def, auto)
lemma st_set_Union_iff [iff]: "st_set(Union(S)) <-> (∀A ∈ S. st_set(A))"
by (simp add: st_set_def, auto)
lemma st_set_Int [intro!]: "st_set(A) | st_set(B) ==> st_set(A Int B)"
by (simp add: st_set_def, auto)
lemma st_set_Inter [intro!]:
"(S=0) | (∃A ∈ S. st_set(A)) ==> st_set(Inter(S))"
apply (simp add: st_set_def Inter_def, auto)
done
lemma st_set_DiffI [intro!]: "st_set(A) ==> st_set(A - B)"
by (simp add: st_set_def, auto)
lemma Collect_Int_state [simp]: "Collect(state,P) Int state = Collect(state,P)"
by auto
lemma state_Int_Collect [simp]: "state Int Collect(state,P) = Collect(state,P)"
by auto
lemma st_setI: "A <= state ==> st_set(A)"
by (simp add: st_set_def)
lemma st_setD: "st_set(A) ==> A<=state"
by (simp add: st_set_def)
lemma st_set_subset: "[| st_set(A); B<=A |] ==> st_set(B)"
by (simp add: st_set_def, auto)
lemma state_update_type:
"[| s ∈ state; x ∈ var; y ∈ type_of(x) |] ==> s(x:=y):state"
apply (simp add: state_def)
apply (blast intro: update_type)
done
lemma st_set_compl [simp]: "st_set(st_compl(A))"
by (simp add: st_compl_def, auto)
lemma st_compl_iff [simp]: "x ∈ st_compl(A) <-> x ∈ state & x ∉ A"
by (simp add: st_compl_def)
lemma st_compl_Collect [simp]:
"st_compl({s ∈ state. P(s)}) = {s ∈ state. ~P(s)}"
by (simp add: st_compl_def, auto)
lemma UN_conj_eq:
"∀d∈D. f(d) ∈ A ==> (\<Union>k∈A. {d∈D. P(d) & f(d) = k}) = {d∈D. P(d)}"
by blast
end