--- a/src/HOL/Word/TdThs.thy Wed Jun 30 21:29:58 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,228 +0,0 @@
-(*
- 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 \<Longrightarrow> \<forall>x. Rep x \<in> A" and
- tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x" and
- tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> 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 \<in> A ==> y \<in> 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 \<in> A) = (x \<in> 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 \<in> A) = (x \<in> 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)
-
-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 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: "\<And>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
-