# Theory Misc_Typedef

theory Misc_Typedef
imports Main
```(*
Author:     Jeremy Dawson and Gerwin Klein, NICTA

Consequences of type definition theorems, and of extended type definition.
*)

section ‹Type Definition Theorems›

theory Misc_Typedef
imports Main
begin

section "More lemmas about normal type definitions"

lemma tdD1: "type_definition Rep Abs A ⟹ ∀x. Rep x ∈ A"
and tdD2: "type_definition Rep Abs A ⟹ ∀x. Abs (Rep x) = x"
and tdD3: "type_definition Rep Abs A ⟹ ∀y. y ∈ A ⟶ Rep (Abs y) = y"
by (auto simp: type_definition_def)

lemma td_nat_int: "type_definition int nat (Collect ((≤) 0))"
unfolding type_definition_def by auto

context type_definition
begin

declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]

lemma Abs_eqD: "Abs x = Abs y ⟹ x ∈ A ⟹ y ∈ A ⟹ x = y"

lemma Abs_inverse': "r ∈ A ⟹ Abs r = a ⟹ Rep a = r"
by (safe elim!: Abs_inverse)

lemma Rep_comp_inverse: "Rep ∘ f = g ⟹ Abs ∘ g = f"
using Rep_inverse by auto

lemma Rep_eqD [elim!]: "Rep x = Rep y ⟹ x = y"
by simp

lemma Rep_inverse': "Rep a = r ⟹ Abs r = a"
by (safe intro!: Rep_inverse)

lemma comp_Abs_inverse: "f ∘ Abs = g ⟹ g ∘ Rep = f"
using Rep_inverse by auto

lemma set_Rep: "A = range Rep"
proof (rule set_eqI)
show "x ∈ A ⟷ x ∈ range Rep" for x
by (auto dest: Abs_inverse [of x, symmetric])
qed

lemma set_Rep_Abs: "A = range (Rep ∘ Abs)"
proof (rule set_eqI)
show "x ∈ A ⟷ x ∈ range (Rep ∘ Abs)" for x
by (auto dest: Abs_inverse [of x, symmetric])
qed

lemma Abs_inj_on: "inj_on Abs A"
unfolding inj_on_def
by (auto dest: Abs_inject [THEN iffD1])

lemma image: "Abs ` A = UNIV"
by (auto intro!: image_eqI)

lemmas td_thm = type_definition_axioms

lemma fns1: "Rep ∘ fa = fr ∘ Rep ∨ fa ∘ Abs = Abs ∘ fr ⟹ Abs ∘ fr ∘ Rep = fa"
by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)

lemmas fns1a = disjI1 [THEN fns1]
lemmas fns1b = disjI2 [THEN fns1]

lemma fns4: "Rep ∘ fa ∘ Abs = fr ⟹ Rep ∘ fa = fr ∘ Rep ∧ fa ∘ Abs = Abs ∘ fr"
by auto

end

interpretation nat_int: type_definition int nat "Collect ((≤) 0)"
by (rule td_nat_int)

declare
nat_int.Rep_cases [cases del]
nat_int.Abs_cases [cases del]
nat_int.Rep_induct [induct del]
nat_int.Abs_induct [induct del]

subsection "Extended form of type definition predicate"

lemma td_conds:
"norm ∘ norm = norm ⟹
fr ∘ norm = norm ∘ fr ⟷ norm ∘ fr ∘ norm = fr ∘ norm ∧ norm ∘ fr ∘ norm = norm ∘ fr"
apply safe
done

lemma fn_comm_power: "fa ∘ tr = tr ∘ fr ⟹ fa ^^ n ∘ tr = tr ∘ fr ^^ n"
apply (rule ext)
apply (induct n)
apply (auto dest: fun_cong)
done

lemmas fn_comm_power' =
ext [THEN fn_comm_power, THEN fun_cong, unfolded o_def]

locale td_ext = type_definition +
fixes norm
assumes eq_norm: "⋀x. Rep (Abs x) = norm x"
begin

lemma Abs_norm [simp]: "Abs (norm x) = Abs x"
using eq_norm [of x] by (auto elim: Rep_inverse')

lemma td_th: "g ∘ Abs = f ⟹ f (Rep x) = g x"
by (drule comp_Abs_inverse [symmetric]) simp

lemma eq_norm': "Rep ∘ Abs = norm"
by (auto simp: eq_norm)

lemma norm_Rep [simp]: "norm (Rep x) = Rep x"
by (auto simp: eq_norm' intro: td_th)

lemmas td = td_thm

lemma set_iff_norm: "w ∈ A ⟷ w = norm w"
by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])

lemma inverse_norm: "Abs n = w ⟷ Rep w = norm n"
apply (rule iffI)
done

lemma norm_eq_iff: "norm x = norm y ⟷ Abs x = Abs y"

lemma norm_comps:
"Abs ∘ norm = Abs"
"norm ∘ Rep = Rep"
"norm ∘ norm = norm"
by (auto simp: eq_norm' [symmetric] o_def)

lemmas norm_norm [simp] = norm_comps

lemma fns5: "Rep ∘ fa ∘ Abs = fr ⟹ fr ∘ norm = fr ∧ norm ∘ fr = fr"
by (fold eq_norm') auto

text ‹
following give conditions for converses to ‹td_fns1›
▪ the condition ‹norm ∘ fr ∘ norm = fr ∘ norm› says that
‹fr› takes normalised arguments to normalised results
▪ ‹norm ∘ fr ∘ norm = norm ∘ fr› says that ‹fr›
takes norm-equivalent arguments to norm-equivalent results
▪ ‹fr ∘ norm = fr› says that ‹fr›
takes norm-equivalent arguments to the same result
▪ ‹norm ∘ fr = fr› says that ‹fr› takes any argument to a normalised result
›
lemma fns2: "Abs ∘ fr ∘ Rep = fa ⟹ norm ∘ fr ∘ norm = fr ∘ norm ⟷ Rep ∘ fa = fr ∘ Rep"
apply (fold eq_norm')
apply safe
prefer 2
apply (rule ext)
apply (drule_tac x="Rep x" in fun_cong)
apply auto
done

lemma fns3: "Abs ∘ fr ∘ Rep = fa ⟹ norm ∘ fr ∘ norm = norm ∘ fr ⟷ fa ∘ Abs = Abs ∘ fr"
apply (fold eq_norm')
apply safe
prefer 2
apply (rule ext)
apply (drule_tac f="a ∘ b" for a b in fun_cong)
apply simp
done

lemma fns: "fr ∘ norm = norm ∘ fr ⟹ fa ∘ Abs = Abs ∘ fr ⟷ Rep ∘ fa = fr ∘ Rep"
apply safe
apply (frule fns1b)
prefer 2
apply (frule fns1a)
apply (rule fns3 [THEN iffD1])
prefer 3
apply (rule fns2 [THEN iffD1])