(* ID: $Id: TdThs.thy,v 1.4 2008/04/04 11:40:27 haftmann Exp $ Author: Jeremy Dawson and Gerwin Klein, NICTA consequences of type definition theorems, and of extended type definition theorems *) header {* Type Definition Theorems *} theory TdThs 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 (op <= 0))" unfolding type_definition_def by auto context type_definition begin lemmas Rep' [iff] = Rep [simplified] (* if A is given as Collect .. *) declare Rep_inverse [simp] Rep_inject [simp] lemma Abs_eqD: "Abs x = Abs y ==> x ∈ A ==> y ∈ A ==> x = y" by (simp add: Abs_inject) lemma Abs_inverse': "r : A ==> Abs r = a ==> Rep a = r" by (safe elim!: Abs_inverse) lemma Rep_comp_inverse: "Rep o f = g ==> Abs o g = f" using Rep_inverse by (auto intro: ext) 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 o Abs = g ==> g o Rep = f" using Rep_inverse by (auto intro: ext) lemma set_Rep: "A = range Rep" proof (rule set_ext) fix x show "(x ∈ A) = (x ∈ range Rep)" by (auto dest: Abs_inverse [of x, symmetric]) qed lemma set_Rep_Abs: "A = range (Rep o Abs)" proof (rule set_ext) fix x show "(x ∈ A) = (x ∈ range (Rep o Abs))" 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 o fa = fr o Rep | fa o Abs = Abs o fr ==> Abs o fr o 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 o fa o Abs = fr ==> Rep o fa = fr o Rep & fa o Abs = Abs o fr" by (auto intro!: ext) end interpretation nat_int: type_definition [int nat "Collect (op <= 0)"] by (rule td_nat_int) -- "resetting to the default nat induct and cases rules" declare Nat.induct [case_names 0 Suc, induct type] declare Nat.exhaust [case_names 0 Suc, cases type] subsection "Extended form of type definition predicate" lemma td_conds: "norm o norm = norm ==> (fr o norm = norm o fr) = (norm o fr o norm = fr o norm & norm o fr o norm = norm o fr)" apply safe apply (simp_all add: o_assoc [symmetric]) apply (simp_all add: o_assoc) done lemma fn_comm_power: "fa o tr = tr o fr ==> fa ^ n o tr = tr o 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, standard] 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 o Abs = f ==> f (Rep x) = g x" by (drule comp_Abs_inverse [symmetric]) simp lemma eq_norm': "Rep o Abs = norm" by (auto simp: eq_norm intro!: ext) 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) apply (clarsimp simp add: eq_norm) apply (simp add: eq_norm' [symmetric]) done lemma norm_eq_iff: "(norm x = norm y) = (Abs x = Abs y)" by (simp add: eq_norm' [symmetric]) lemma norm_comps: "Abs o norm = Abs" "norm o Rep = Rep" "norm o norm = norm" by (auto simp: eq_norm' [symmetric] o_def) lemmas norm_norm [simp] = norm_comps lemma fns5: "Rep o fa o Abs = fr ==> fr o norm = fr & norm o fr = fr" by (fold eq_norm') (auto intro!: ext) (* following give conditions for converses to td_fns1 the condition (norm o fr o norm = fr o norm) says that fr takes normalised arguments to normalised results, (norm o fr o norm = norm o fr) says that fr takes norm-equivalent arguments to norm-equivalent results, (fr o norm = fr) says that fr takes norm-equivalent arguments to the same result, and (norm o fr = fr) says that fr takes any argument to a normalised result *) lemma fns2: "Abs o fr o Rep = fa ==> (norm o fr o norm = fr o norm) = (Rep o fa = fr o Rep)" apply (fold eq_norm') apply safe prefer 2 apply (simp add: o_assoc) apply (rule ext) apply (drule_tac x="Rep x" in fun_cong) apply auto done lemma fns3: "Abs o fr o Rep = fa ==> (norm o fr o norm = norm o fr) = (fa o Abs = Abs o fr)" apply (fold eq_norm') apply safe prefer 2 apply (simp add: o_assoc [symmetric]) apply (rule ext) apply (drule fun_cong) apply simp done lemma fns: "fr o norm = norm o fr ==> (fa o Abs = Abs o fr) = (Rep o fa = fr o Rep)" apply safe apply (frule fns1b) prefer 2 apply (frule fns1a) apply (rule fns3 [THEN iffD1]) prefer 3 apply (rule fns2 [THEN iffD1]) apply (simp_all add: o_assoc [symmetric]) apply (simp_all add: o_assoc) done lemma range_norm: "range (Rep o Abs) = A" by (simp add: set_Rep_Abs) end lemmas td_ext_def' = td_ext_def [unfolded type_definition_def td_ext_axioms_def] end
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
lemma td_nat_int:
type_definition int nat (Collect (op ≤ 0))
lemma Rep':
Rep x ∈ A
lemma Abs_eqD:
[| Abs x = Abs y; x ∈ A; y ∈ A |] ==> x = y
lemma Abs_inverse':
[| r ∈ A; Abs r = a |] ==> Rep a = r
lemma Rep_comp_inverse:
Rep o f = g ==> Abs o g = f
lemma Rep_eqD:
Rep x = Rep y ==> x = y
lemma Rep_inverse':
Rep a = r ==> Abs r = a
lemma comp_Abs_inverse:
f o Abs = g ==> g o Rep = f
lemma set_Rep:
A = range Rep
lemma set_Rep_Abs:
A = range (Rep o Abs)
lemma Abs_inj_on:
inj_on Abs A
lemma image:
Abs ` A = UNIV
lemma td_thm:
type_definition Rep Abs A
lemma fns1:
Rep o fa = fr o Rep ∨ fa o Abs = Abs o fr ==> Abs o fr o Rep = fa
lemma fns1a:
Rep o fa = fr o Rep ==> Abs o fr o Rep = fa
lemma fns1b:
fa o Abs = Abs o fr ==> Abs o fr o Rep = fa
lemma fns4:
Rep o fa o Abs = fr ==> Rep o fa = fr o Rep ∧ fa o Abs = Abs o fr
lemma td_conds:
norm o norm = norm
==> (fr o norm = norm o fr) =
(norm o fr o norm = fr o norm ∧ norm o fr o norm = norm o fr)
lemma fn_comm_power:
fa o tr = tr o fr ==> fa ^ n o tr = tr o fr ^ n
lemma fn_comm_power':
(!!x. fa (tr x) = tr (fr x)) ==> (fa ^ n) (tr x) = tr ((fr ^ n) x)
lemma Abs_norm:
Abs (norm x) = Abs x
lemma td_th:
g o Abs = f ==> f (Rep x) = g x
lemma eq_norm':
Rep o Abs = norm
lemma norm_Rep:
norm (Rep x) = Rep x
lemma td:
type_definition Rep Abs A
lemma set_iff_norm:
(w ∈ A) = (w = norm w)
lemma inverse_norm:
(Abs n = w) = (Rep w = norm n)
lemma norm_eq_iff:
(norm x = norm y) = (Abs x = Abs y)
lemma norm_comps:
Abs o norm = Abs
norm o Rep = Rep
norm o norm = norm
lemma norm_norm:
Abs o norm = Abs
norm o Rep = Rep
norm o norm = norm
lemma fns5:
Rep o fa o Abs = fr ==> fr o norm = fr ∧ norm o fr = fr
lemma fns2:
Abs o fr o Rep = fa ==> (norm o fr o norm = fr o norm) = (Rep o fa = fr o Rep)
lemma fns3:
Abs o fr o Rep = fa ==> (norm o fr o norm = norm o fr) = (fa o Abs = Abs o fr)
lemma fns:
fr o norm = norm o fr ==> (fa o Abs = Abs o fr) = (Rep o fa = fr o Rep)
lemma range_norm:
range (Rep o Abs) = A
lemma td_ext_def':
td_ext Rep Abs A norm ==
((∀x. Rep x ∈ A) ∧ (∀x. Abs (Rep x) = x) ∧ (∀y. y ∈ A --> Rep (Abs y) = y)) ∧
(∀x. Rep (Abs x) = norm x)