--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/QuotBase.thy Mon Jan 25 18:13:44 2010 +0100
@@ -0,0 +1,694 @@
+(* Title: QuotBase.thy
+ Author: Cezary Kaliszyk and Christian Urban
+*)
+
+theory QuotBase
+imports Plain ATP_Linkup Predicate
+begin
+
+text {*
+ Basic definition for equivalence relations
+ that are represented by predicates.
+*}
+
+definition
+ "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
+
+definition
+ "reflp E \<equiv> \<forall>x. E x x"
+
+definition
+ "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
+
+definition
+ "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
+
+lemma equivp_reflp_symp_transp:
+ shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
+ unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq
+ by blast
+
+lemma equivp_reflp:
+ shows "equivp E \<Longrightarrow> (\<And>x. E x x)"
+ by (simp only: equivp_reflp_symp_transp reflp_def)
+
+lemma equivp_symp:
+ shows "equivp E \<Longrightarrow> (\<And>x y. E x y \<Longrightarrow> E y x)"
+ by (metis equivp_reflp_symp_transp symp_def)
+
+lemma equivp_transp:
+ shows "equivp E \<Longrightarrow> (\<And>x y z. E x y \<Longrightarrow> E y z \<Longrightarrow> E x z)"
+ by (metis equivp_reflp_symp_transp transp_def)
+
+lemma equivpI:
+ assumes "reflp R" "symp R" "transp R"
+ shows "equivp R"
+ using assms by (simp add: equivp_reflp_symp_transp)
+
+lemma eq_imp_rel:
+ shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"
+by (simp add: equivp_reflp)
+
+text {* Partial equivalences: not yet used anywhere *}
+definition
+ "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
+
+lemma equivp_IMP_part_equivp:
+ assumes a: "equivp E"
+ shows "part_equivp E"
+ using a
+ unfolding equivp_def part_equivp_def
+ by auto
+
+text {* Composition of Relations *}
+abbreviation
+ rel_conj (infixr "OOO" 75)
+where
+ "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
+
+section {* Quotient Predicate *}
+
+definition
+ "Quotient E Abs Rep \<equiv>
+ (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
+ (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
+
+lemma Quotient_abs_rep:
+ assumes a: "Quotient E Abs Rep"
+ shows "Abs (Rep a) = a"
+ using a
+ unfolding Quotient_def
+ by simp
+
+lemma Quotient_rep_reflp:
+ assumes a: "Quotient E Abs Rep"
+ shows "E (Rep a) (Rep a)"
+ using a
+ unfolding Quotient_def
+ by blast
+
+lemma Quotient_rel:
+ assumes a: "Quotient E Abs Rep"
+ shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
+ using a
+ unfolding Quotient_def
+ by blast
+
+lemma Quotient_rel_rep:
+ assumes a: "Quotient R Abs Rep"
+ shows "R (Rep a) (Rep b) = (a = b)"
+ using a
+ unfolding Quotient_def
+ by metis
+
+lemma Quotient_rep_abs:
+ assumes a: "Quotient R Abs Rep"
+ shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
+ using a unfolding Quotient_def
+ by blast
+
+lemma Quotient_rel_abs:
+ assumes a: "Quotient E Abs Rep"
+ shows "E r s \<Longrightarrow> Abs r = Abs s"
+ using a unfolding Quotient_def
+ by blast
+
+lemma Quotient_symp:
+ assumes a: "Quotient E Abs Rep"
+ shows "symp E"
+ using a unfolding Quotient_def symp_def
+ by metis
+
+lemma Quotient_transp:
+ assumes a: "Quotient E Abs Rep"
+ shows "transp E"
+ using a unfolding Quotient_def transp_def
+ by metis
+
+section {* Lemmas about (op =) *}
+
+lemma identity_equivp:
+ shows "equivp (op =)"
+ unfolding equivp_def
+ by auto
+
+lemma identity_quotient:
+ shows "Quotient (op =) id id"
+ unfolding Quotient_def id_def
+ by blast
+
+section {* Function map and function relation *}
+
+definition
+ fun_map (infixr "--->" 55)
+where
+[simp]: "fun_map f g h x = g (h (f x))"
+
+definition
+ fun_rel (infixr "===>" 55)
+where
+[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
+
+
+lemma fun_map_id:
+ shows "(id ---> id) = id"
+ by (simp add: expand_fun_eq id_def)
+
+lemma fun_rel_eq:
+ shows "(op =) ===> (op =) \<equiv> (op =)"
+ by (rule eq_reflection) (simp add: expand_fun_eq)
+
+lemma fun_quotient:
+ assumes q1: "Quotient R1 abs1 rep1"
+ and q2: "Quotient R2 abs2 rep2"
+ shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
+proof -
+ have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
+ apply(simp add: expand_fun_eq)
+ using q1 q2
+ apply(simp add: Quotient_def)
+ done
+ moreover
+ have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
+ apply(auto)
+ using q1 q2 unfolding Quotient_def
+ apply(metis)
+ done
+ moreover
+ have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
+ (rep1 ---> abs2) r = (rep1 ---> abs2) s)"
+ apply(auto simp add: expand_fun_eq)
+ using q1 q2 unfolding Quotient_def
+ apply(metis)
+ using q1 q2 unfolding Quotient_def
+ apply(metis)
+ using q1 q2 unfolding Quotient_def
+ apply(metis)
+ using q1 q2 unfolding Quotient_def
+ apply(metis)
+ done
+ ultimately
+ show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
+ unfolding Quotient_def by blast
+qed
+
+section {* Respects predicate *}
+
+definition
+ Respects
+where
+ "Respects R x \<equiv> (R x x)"
+
+lemma in_respects:
+ shows "(x \<in> Respects R) = R x x"
+ unfolding mem_def Respects_def by simp
+
+lemma equals_rsp:
+ assumes q: "Quotient R Abs Rep"
+ and a: "R xa xb" "R ya yb"
+ shows "R xa ya = R xb yb"
+ using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_def
+ using a by blast
+
+lemma lambda_prs:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
+ unfolding expand_fun_eq
+ using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
+ by simp
+
+lemma lambda_prs1:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
+ unfolding expand_fun_eq
+ using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
+ by simp
+
+lemma rep_abs_rsp:
+ assumes q: "Quotient R Abs Rep"
+ and a: "R x1 x2"
+ shows "R x1 (Rep (Abs x2))"
+ using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
+
+lemma rep_abs_rsp_left:
+ assumes q: "Quotient R Abs Rep"
+ and a: "R x1 x2"
+ shows "R (Rep (Abs x1)) x2"
+using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
+
+(* In the following theorem R1 can be instantiated with anything,
+ but we know some of the types of the Rep and Abs functions;
+ so by solving Quotient assumptions we can get a unique R1 that
+ will be provable; which is why we need to use apply_rsp and
+ not the primed version *)
+lemma apply_rsp:
+ fixes f g::"'a \<Rightarrow> 'c"
+ assumes q: "Quotient R1 Abs1 Rep1"
+ and a: "(R1 ===> R2) f g" "R1 x y"
+ shows "R2 (f x) (g y)"
+ using a by simp
+
+lemma apply_rsp':
+ assumes a: "(R1 ===> R2) f g" "R1 x y"
+ shows "R2 (f x) (g y)"
+ using a by simp
+
+(* Set of lemmas for regularisation of ball and bex *)
+
+lemma ball_reg_eqv:
+ fixes P :: "'a \<Rightarrow> bool"
+ assumes a: "equivp R"
+ shows "Ball (Respects R) P = (All P)"
+ by (metis equivp_def in_respects a)
+
+lemma bex_reg_eqv:
+ fixes P :: "'a \<Rightarrow> bool"
+ assumes a: "equivp R"
+ shows "Bex (Respects R) P = (Ex P)"
+ by (metis equivp_def in_respects a)
+
+lemma ball_reg_right:
+ assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
+ shows "All P \<longrightarrow> Ball R Q"
+ by (metis COMBC_def Collect_def Collect_mem_eq a)
+
+lemma bex_reg_left:
+ assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
+ shows "Bex R Q \<longrightarrow> Ex P"
+ by (metis COMBC_def Collect_def Collect_mem_eq a)
+
+lemma ball_reg_left:
+ assumes a: "equivp R"
+ shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
+ by (metis equivp_reflp in_respects a)
+
+lemma bex_reg_right:
+ assumes a: "equivp R"
+ shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
+ by (metis equivp_reflp in_respects a)
+
+lemma ball_reg_eqv_range:
+ fixes P::"'a \<Rightarrow> bool"
+ and x::"'a"
+ assumes a: "equivp R2"
+ shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
+ apply(rule iffI)
+ apply(rule allI)
+ apply(drule_tac x="\<lambda>y. f x" in bspec)
+ apply(simp add: in_respects)
+ apply(rule impI)
+ using a equivp_reflp_symp_transp[of "R2"]
+ apply(simp add: reflp_def)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma bex_reg_eqv_range:
+ assumes a: "equivp R2"
+ shows "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
+ apply(auto)
+ apply(rule_tac x="\<lambda>y. f x" in bexI)
+ apply(simp)
+ apply(simp add: Respects_def in_respects)
+ apply(rule impI)
+ using a equivp_reflp_symp_transp[of "R2"]
+ apply(simp add: reflp_def)
+ done
+
+lemma all_reg:
+ assumes a: "!x :: 'a. (P x --> Q x)"
+ and b: "All P"
+ shows "All Q"
+ using a b by (metis)
+
+lemma ex_reg:
+ assumes a: "!x :: 'a. (P x --> Q x)"
+ and b: "Ex P"
+ shows "Ex Q"
+ using a b by (metis)
+
+lemma ball_reg:
+ assumes a: "!x :: 'a. (R x --> P x --> Q x)"
+ and b: "Ball R P"
+ shows "Ball R Q"
+ using a b by (metis COMBC_def Collect_def Collect_mem_eq)
+
+lemma bex_reg:
+ assumes a: "!x :: 'a. (R x --> P x --> Q x)"
+ and b: "Bex R P"
+ shows "Bex R Q"
+ using a b by (metis COMBC_def Collect_def Collect_mem_eq)
+
+lemma ball_all_comm:
+ "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"
+ by auto
+
+lemma bex_ex_comm:
+ "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"
+ by auto
+
+(* Bounded abstraction *)
+definition
+ Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where
+ "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"
+
+definition
+ Bexeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
+where
+ "Bexeq R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"
+
+(* 3 lemmas needed for proving repabs_inj *)
+lemma ball_rsp:
+ assumes a: "(R ===> (op =)) f g"
+ shows "Ball (Respects R) f = Ball (Respects R) g"
+ using a by (simp add: Ball_def in_respects)
+
+lemma bex_rsp:
+ assumes a: "(R ===> (op =)) f g"
+ shows "(Bex (Respects R) f = Bex (Respects R) g)"
+ using a by (simp add: Bex_def in_respects)
+
+lemma bex1_rsp:
+ assumes a: "(R ===> (op =)) f g"
+ shows "(Bex1 (Respects R) f = Bex1 (Respects R) g)"
+ using a
+by (simp add: Ex1_def Bex1_def in_respects) auto
+
+(* TODO/FIXME: simplify the repetitions in this proof *)
+lemma bexeq_rsp:
+assumes a: "Quotient R absf repf"
+shows "((R ===> op =) ===> op =) (Bexeq R) (Bexeq R)"
+apply simp
+unfolding Bexeq_def
+apply rule
+apply rule
+apply rule
+apply rule
+apply (erule conjE)+
+apply (erule bexE)
+apply rule
+apply (rule_tac x="xa" in bexI)
+apply metis
+apply metis
+apply rule+
+apply (erule_tac x="xb" in ballE)
+prefer 2
+apply (metis)
+apply (erule_tac x="ya" in ballE)
+prefer 2
+apply (metis)
+apply (metis in_respects)
+apply (erule conjE)+
+apply (erule bexE)
+apply rule
+apply (rule_tac x="xa" in bexI)
+apply metis
+apply metis
+apply rule+
+apply (erule_tac x="xb" in ballE)
+prefer 2
+apply (metis)
+apply (erule_tac x="ya" in ballE)
+prefer 2
+apply (metis)
+apply (metis in_respects)
+done
+
+lemma babs_rsp:
+ assumes q: "Quotient R1 Abs1 Rep1"
+ and a: "(R1 ===> R2) f g"
+ shows "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
+ apply (auto simp add: Babs_def)
+ apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
+ using a apply (simp add: Babs_def)
+ apply (simp add: in_respects)
+ using Quotient_rel[OF q]
+ by metis
+
+lemma babs_prs:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ shows "(Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f)) \<equiv> f"
+ apply(rule eq_reflection)
+ apply(rule ext)
+ apply simp
+ apply (subgoal_tac "Rep1 x \<in> Respects R1")
+ apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
+ apply (simp add: in_respects Quotient_rel_rep[OF q1])
+ done
+
+lemma babs_simp:
+ assumes q: "Quotient R1 Abs Rep"
+ shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
+ apply(rule iffI)
+ apply(simp_all only: babs_rsp[OF q])
+ apply(auto simp add: Babs_def)
+ apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
+ apply(metis Babs_def)
+ apply (simp add: in_respects)
+ using Quotient_rel[OF q]
+ by metis
+
+(* If a user proves that a particular functional relation
+ is an equivalence this may be useful in regularising *)
+lemma babs_reg_eqv:
+ shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
+ by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)
+
+(* 3 lemmas needed for cleaning of quantifiers *)
+lemma all_prs:
+ assumes a: "Quotient R absf repf"
+ shows "Ball (Respects R) ((absf ---> id) f) = All f"
+ using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
+by metis
+
+lemma ex_prs:
+ assumes a: "Quotient R absf repf"
+ shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
+ using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
+ by metis
+
+lemma ex1_prs:
+ assumes a: "Quotient R absf repf"
+ shows "((absf ---> id) ---> id) (Bexeq R) f = Ex1 f"
+apply simp
+apply (subst Bexeq_def)
+apply (subst Bex_def)
+apply (subst Ex1_def)
+apply simp
+apply rule
+ apply (erule conjE)+
+ apply (erule_tac exE)
+ apply (erule conjE)
+ apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
+ apply (rule_tac x="absf x" in exI)
+ apply (thin_tac "\<forall>x\<in>Respects R. \<forall>y\<in>Respects R. f (absf x) \<and> f (absf y) \<longrightarrow> R x y")
+ apply (simp)
+ apply rule+
+ using a unfolding Quotient_def
+ apply metis
+ apply rule+
+ apply (erule_tac x="x" in ballE)
+ apply (erule_tac x="y" in ballE)
+ apply simp
+ apply (simp add: in_respects)
+ apply (simp add: in_respects)
+apply (erule_tac exE)
+ apply rule
+ apply (rule_tac x="repf x" in exI)
+ apply (simp only: in_respects)
+ apply rule
+ apply (metis Quotient_rel_rep[OF a])
+using a unfolding Quotient_def apply (simp)
+apply rule+
+using a unfolding Quotient_def in_respects
+apply metis
+done
+
+lemma fun_rel_id:
+ assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
+ shows "(R1 ===> R2) f g"
+ using a by simp
+
+lemma fun_rel_id_asm:
+ assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"
+ shows "A \<longrightarrow> (R1 ===> R2) f g"
+ using a by auto
+
+lemma quot_rel_rsp:
+ assumes a: "Quotient R Abs Rep"
+ shows "(R ===> R ===> op =) R R"
+ apply(rule fun_rel_id)+
+ apply(rule equals_rsp[OF a])
+ apply(assumption)+
+ done
+
+lemma o_prs:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ and q3: "Quotient R3 Abs3 Rep3"
+ shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g"
+ using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
+ unfolding o_def expand_fun_eq by simp
+
+lemma o_rsp:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ and q3: "Quotient R3 Abs3 Rep3"
+ and a1: "(R2 ===> R3) f1 f2"
+ and a2: "(R1 ===> R2) g1 g2"
+ shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
+ using a1 a2 unfolding o_def expand_fun_eq
+ by (auto)
+
+lemma cond_prs:
+ assumes a: "Quotient R absf repf"
+ shows "absf (if a then repf b else repf c) = (if a then b else c)"
+ using a unfolding Quotient_def by auto
+
+lemma if_prs:
+ assumes q: "Quotient R Abs Rep"
+ shows "Abs (If a (Rep b) (Rep c)) = If a b c"
+using Quotient_abs_rep[OF q] by auto
+
+(* q not used *)
+lemma if_rsp:
+ assumes q: "Quotient R Abs Rep"
+ and a: "a1 = a2" "R b1 b2" "R c1 c2"
+ shows "R (If a1 b1 c1) (If a2 b2 c2)"
+using a by auto
+
+lemma let_prs:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f"
+ using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto
+
+lemma let_rsp:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and a1: "(R1 ===> R2) f g"
+ and a2: "R1 x y"
+ shows "R2 ((Let x f)::'c) ((Let y g)::'c)"
+ using apply_rsp[OF q1 a1] a2 by auto
+
+
+
+
+(******************************************)
+(* REST OF THE FILE IS UNUSED (until now) *)
+(******************************************)
+
+lemma in_fun:
+ shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"
+ by (simp add: mem_def)
+
+lemma respects_thm:
+ shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"
+ unfolding Respects_def
+ by (simp add: expand_fun_eq)
+
+lemma respects_rep_abs:
+ assumes a: "Quotient R1 Abs1 Rep1"
+ and b: "Respects (R1 ===> R2) f"
+ and c: "R1 x x"
+ shows "R2 (f (Rep1 (Abs1 x))) (f x)"
+ using a b[simplified respects_thm] c unfolding Quotient_def
+ by blast
+
+lemma respects_mp:
+ assumes a: "Respects (R1 ===> R2) f"
+ and b: "R1 x y"
+ shows "R2 (f x) (f y)"
+ using a b unfolding Respects_def
+ by simp
+
+lemma respects_o:
+ assumes a: "Respects (R2 ===> R3) f"
+ and b: "Respects (R1 ===> R2) g"
+ shows "Respects (R1 ===> R3) (f o g)"
+ using a b unfolding Respects_def
+ by simp
+
+lemma abs_o_rep:
+ assumes a: "Quotient r absf repf"
+ shows "absf o repf = id"
+ apply(rule ext)
+ apply(simp add: Quotient_abs_rep[OF a])
+ done
+
+lemma eq_comp_r: "op = OO R OO op = \<equiv> R"
+ apply (rule eq_reflection)
+ apply (rule ext)+
+ apply auto
+ done
+
+lemma fun_rel_eq_rel:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g)
+ \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"
+ using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq
+ by blast
+
+lemma let_babs:
+ "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam"
+ by (simp add: Babs_def)
+
+lemma fun_rel_equals:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ and r1: "Respects (R1 ===> R2) f"
+ and r2: "Respects (R1 ===> R2) g"
+ shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
+ apply(rule_tac iffI)
+ apply(rule)+
+ apply (rule apply_rsp'[of "R1" "R2"])
+ apply(subst Quotient_rel[OF fun_quotient[OF q1 q2]])
+ apply auto
+ using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
+ apply (metis let_rsp q1)
+ apply (metis fun_rel_eq_rel let_rsp q1 q2 r2)
+ using r1 unfolding Respects_def expand_fun_eq
+ apply(simp (no_asm_use))
+ apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1])
+ done
+
+(* ask Peter: fun_rel_IMP used twice *)
+lemma fun_rel_IMP2:
+ assumes q1: "Quotient R1 Abs1 Rep1"
+ and q2: "Quotient R2 Abs2 Rep2"
+ and r1: "Respects (R1 ===> R2) f"
+ and r2: "Respects (R1 ===> R2) g"
+ and a: "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"
+ shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"
+ using q1 q2 r1 r2 a
+ by (simp add: fun_rel_equals)
+
+lemma lambda_rep_abs_rsp:
+ assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"
+ and r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"
+ shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"
+ using r1 r2 by auto
+
+(* ask peter what are literal_case *)
+(* literal_case_PRS *)
+(* literal_case_RSP *)
+(* Cez: !f x. literal_case f x = f x *)
+
+(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
+lemma id_prs:
+ assumes q: "Quotient R Abs Rep"
+ shows "Abs (id (Rep e)) = id e"
+ using Quotient_abs_rep[OF q] by auto
+
+lemma id_rsp:
+ assumes q: "Quotient R Abs Rep"
+ and a: "R e1 e2"
+ shows "R (id e1) (id e2)"
+ using a by auto
+
+end
+
--- a/Quot/QuotMain.thy Mon Jan 25 17:53:08 2010 +0100
+++ b/Quot/QuotMain.thy Mon Jan 25 18:13:44 2010 +0100
@@ -3,7 +3,7 @@
*)
theory QuotMain
-imports QuotScript Prove
+imports QuotBase
uses ("quotient_info.ML")
("quotient_typ.ML")
("quotient_def.ML")
@@ -61,11 +61,6 @@
show "Abs (R (Eps (Rep a))) = a" by simp
qed
-lemma rep_refl:
- shows "R (rep a) (rep a)"
-unfolding rep_def
-by (simp add: equivp[simplified equivp_def])
-
lemma lem7:
shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")
proof -
@@ -79,6 +74,12 @@
unfolding abs_def
by (simp only: equivp[simplified equivp_def] lem7)
+lemma rep_refl:
+ shows "R (rep a) (rep a)"
+unfolding rep_def
+by (simp add: equivp[simplified equivp_def])
+
+
lemma rep_abs_rsp:
shows "R f (rep (abs g)) = R f g"
and "R (rep (abs g)) f = R g f"
--- a/Quot/QuotScript.thy Mon Jan 25 17:53:08 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,682 +0,0 @@
-theory QuotScript
-imports Plain ATP_Linkup Predicate
-begin
-
-definition
- "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
-
-definition
- "reflp E \<equiv> \<forall>x. E x x"
-
-definition
- "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
-
-definition
- "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
-
-lemma equivp_reflp_symp_transp:
- shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
- unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq
- by (blast)
-
-lemma equivp_reflp:
- shows "equivp E \<Longrightarrow> (\<And>x. E x x)"
- by (simp only: equivp_reflp_symp_transp reflp_def)
-
-lemma equivp_symp:
- shows "equivp E \<Longrightarrow> (\<And>x y. E x y \<Longrightarrow> E y x)"
- by (metis equivp_reflp_symp_transp symp_def)
-
-lemma equivp_transp:
- shows "equivp E \<Longrightarrow> (\<And>x y z. E x y \<Longrightarrow> E y z \<Longrightarrow> E x z)"
- by (metis equivp_reflp_symp_transp transp_def)
-
-lemma equivpI:
- assumes "reflp R" "symp R" "transp R"
- shows "equivp R"
- using assms by (simp add: equivp_reflp_symp_transp)
-
-lemma eq_imp_rel:
- shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"
-by (simp add: equivp_reflp)
-
-definition
- "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
-
-lemma equivp_IMP_part_equivp:
- assumes a: "equivp E"
- shows "part_equivp E"
- using a unfolding equivp_def part_equivp_def
- by auto
-
-
-abbreviation
- rel_conj (infixr "OOO" 75)
-where
- "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
-
-definition
- "Quotient E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and>
- (\<forall>a. E (Rep a) (Rep a)) \<and>
- (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
-
-(* TEST
-lemma
- fixes Abs1::"'b \<Rightarrow> 'c"
- and Abs2::"'a \<Rightarrow> 'b"
- and Rep1::"'c \<Rightarrow> 'b"
- and Rep2::"'b \<Rightarrow> 'a"
- assumes "Quotient R1 Abs1 Rep1"
- and "Quotient R2 Abs2 Rep2"
- shows "Quotient (f R2 R1) (Abs1 \<circ> Abs2) (Rep2 \<circ> Rep1)"
-*)
-
-lemma Quotient_abs_rep:
- assumes a: "Quotient E Abs Rep"
- shows "Abs (Rep a) \<equiv> a"
- using a unfolding Quotient_def
- by simp
-
-lemma Quotient_rep_reflp:
- assumes a: "Quotient E Abs Rep"
- shows "E (Rep a) (Rep a)"
- using a unfolding Quotient_def
- by blast
-
-lemma Quotient_rel:
- assumes a: "Quotient E Abs Rep"
- shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
- using a unfolding Quotient_def
- by blast
-
-lemma Quotient_rel_rep:
- assumes a: "Quotient R Abs Rep"
- shows "R (Rep a) (Rep b) \<equiv> (a = b)"
- apply (rule eq_reflection)
- using a unfolding Quotient_def
- by metis
-
-lemma Quotient_rep_abs:
- assumes a: "Quotient R Abs Rep"
- shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
- using a unfolding Quotient_def
- by blast
-
-lemma Quotient_rel_abs:
- assumes a: "Quotient E Abs Rep"
- shows "E r s \<Longrightarrow> Abs r = Abs s"
- using a unfolding Quotient_def
- by blast
-
-lemma identity_equivp:
- shows "equivp (op =)"
- unfolding equivp_def
- by auto
-
-lemma identity_quotient:
- shows "Quotient (op =) id id"
- unfolding Quotient_def id_def
- by blast
-
-lemma Quotient_symp:
- assumes a: "Quotient E Abs Rep"
- shows "symp E"
- using a unfolding Quotient_def symp_def
- by metis
-
-lemma Quotient_transp:
- assumes a: "Quotient E Abs Rep"
- shows "transp E"
- using a unfolding Quotient_def transp_def
- by metis
-
-definition
- fun_map (infixr "--->" 55)
-where
-[simp]: "fun_map f g h x = g (h (f x))"
-
-lemma fun_map_id:
- shows "(id ---> id) = id"
- by (simp add: expand_fun_eq id_def)
-
-definition
- fun_rel (infixr "===>" 55)
-where
-[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
-
-lemma fun_rel_eq:
- "(op =) ===> (op =) \<equiv> (op =)"
- by (rule eq_reflection) (simp add: expand_fun_eq)
-
-lemma fun_quotient:
- assumes q1: "Quotient R1 abs1 rep1"
- and q2: "Quotient R2 abs2 rep2"
- shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
-proof -
- have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
- apply(simp add: expand_fun_eq)
- using q1 q2
- apply(simp add: Quotient_def)
- done
- moreover
- have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
- apply(auto)
- using q1 q2 unfolding Quotient_def
- apply(metis)
- done
- moreover
- have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
- (rep1 ---> abs2) r = (rep1 ---> abs2) s)"
- apply(auto simp add: expand_fun_eq)
- using q1 q2 unfolding Quotient_def
- apply(metis)
- using q1 q2 unfolding Quotient_def
- apply(metis)
- using q1 q2 unfolding Quotient_def
- apply(metis)
- using q1 q2 unfolding Quotient_def
- apply(metis)
- done
- ultimately
- show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
- unfolding Quotient_def by blast
-qed
-
-definition
- Respects
-where
- "Respects R x \<equiv> (R x x)"
-
-lemma in_respects:
- shows "(x \<in> Respects R) = R x x"
- unfolding mem_def Respects_def by simp
-
-lemma equals_rsp:
- assumes q: "Quotient R Abs Rep"
- and a: "R xa xb" "R ya yb"
- shows "R xa ya = R xb yb"
- using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_def
- using a by blast
-
-lemma lambda_prs:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
- unfolding expand_fun_eq
- using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
- by simp
-
-lemma lambda_prs1:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
- unfolding expand_fun_eq
- using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
- by simp
-
-lemma rep_abs_rsp:
- assumes q: "Quotient R Abs Rep"
- and a: "R x1 x2"
- shows "R x1 (Rep (Abs x2))"
- using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
-
-lemma rep_abs_rsp_left:
- assumes q: "Quotient R Abs Rep"
- and a: "R x1 x2"
- shows "R (Rep (Abs x1)) x2"
-using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
-
-(* In the following theorem R1 can be instantiated with anything,
- but we know some of the types of the Rep and Abs functions;
- so by solving Quotient assumptions we can get a unique R1 that
- will be provable; which is why we need to use apply_rsp and
- not the primed version *)
-lemma apply_rsp:
- fixes f g::"'a \<Rightarrow> 'c"
- assumes q: "Quotient R1 Abs1 Rep1"
- and a: "(R1 ===> R2) f g" "R1 x y"
- shows "R2 (f x) (g y)"
- using a by simp
-
-lemma apply_rsp':
- assumes a: "(R1 ===> R2) f g" "R1 x y"
- shows "R2 (f x) (g y)"
- using a by simp
-
-(* Set of lemmas for regularisation of ball and bex *)
-
-lemma ball_reg_eqv:
- fixes P :: "'a \<Rightarrow> bool"
- assumes a: "equivp R"
- shows "Ball (Respects R) P = (All P)"
- by (metis equivp_def in_respects a)
-
-lemma bex_reg_eqv:
- fixes P :: "'a \<Rightarrow> bool"
- assumes a: "equivp R"
- shows "Bex (Respects R) P = (Ex P)"
- by (metis equivp_def in_respects a)
-
-lemma ball_reg_right:
- assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
- shows "All P \<longrightarrow> Ball R Q"
- by (metis COMBC_def Collect_def Collect_mem_eq a)
-
-lemma bex_reg_left:
- assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
- shows "Bex R Q \<longrightarrow> Ex P"
- by (metis COMBC_def Collect_def Collect_mem_eq a)
-
-lemma ball_reg_left:
- assumes a: "equivp R"
- shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
- by (metis equivp_reflp in_respects a)
-
-lemma bex_reg_right:
- assumes a: "equivp R"
- shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
- by (metis equivp_reflp in_respects a)
-
-lemma ball_reg_eqv_range:
- fixes P::"'a \<Rightarrow> bool"
- and x::"'a"
- assumes a: "equivp R2"
- shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
- apply(rule iffI)
- apply(rule allI)
- apply(drule_tac x="\<lambda>y. f x" in bspec)
- apply(simp add: in_respects)
- apply(rule impI)
- using a equivp_reflp_symp_transp[of "R2"]
- apply(simp add: reflp_def)
- apply(simp)
- apply(simp)
- done
-
-lemma bex_reg_eqv_range:
- assumes a: "equivp R2"
- shows "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
- apply(auto)
- apply(rule_tac x="\<lambda>y. f x" in bexI)
- apply(simp)
- apply(simp add: Respects_def in_respects)
- apply(rule impI)
- using a equivp_reflp_symp_transp[of "R2"]
- apply(simp add: reflp_def)
- done
-
-lemma all_reg:
- assumes a: "!x :: 'a. (P x --> Q x)"
- and b: "All P"
- shows "All Q"
- using a b by (metis)
-
-lemma ex_reg:
- assumes a: "!x :: 'a. (P x --> Q x)"
- and b: "Ex P"
- shows "Ex Q"
- using a b by (metis)
-
-lemma ball_reg:
- assumes a: "!x :: 'a. (R x --> P x --> Q x)"
- and b: "Ball R P"
- shows "Ball R Q"
- using a b by (metis COMBC_def Collect_def Collect_mem_eq)
-
-lemma bex_reg:
- assumes a: "!x :: 'a. (R x --> P x --> Q x)"
- and b: "Bex R P"
- shows "Bex R Q"
- using a b by (metis COMBC_def Collect_def Collect_mem_eq)
-
-lemma ball_all_comm:
- "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"
- by auto
-
-lemma bex_ex_comm:
- "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"
- by auto
-
-(* Bounded abstraction *)
-definition
- Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
-where
- "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"
-
-definition
- Bexeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
-where
- "Bexeq R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"
-
-(* 3 lemmas needed for proving repabs_inj *)
-lemma ball_rsp:
- assumes a: "(R ===> (op =)) f g"
- shows "Ball (Respects R) f = Ball (Respects R) g"
- using a by (simp add: Ball_def in_respects)
-
-lemma bex_rsp:
- assumes a: "(R ===> (op =)) f g"
- shows "(Bex (Respects R) f = Bex (Respects R) g)"
- using a by (simp add: Bex_def in_respects)
-
-lemma bex1_rsp:
- assumes a: "(R ===> (op =)) f g"
- shows "(Bex1 (Respects R) f = Bex1 (Respects R) g)"
- using a
-by (simp add: Ex1_def Bex1_def in_respects) auto
-
-(* TODO/FIXME: simplify the repetitions in this proof *)
-lemma bexeq_rsp:
-assumes a: "Quotient R absf repf"
-shows "((R ===> op =) ===> op =) (Bexeq R) (Bexeq R)"
-apply simp
-unfolding Bexeq_def
-apply rule
-apply rule
-apply rule
-apply rule
-apply (erule conjE)+
-apply (erule bexE)
-apply rule
-apply (rule_tac x="xa" in bexI)
-apply metis
-apply metis
-apply rule+
-apply (erule_tac x="xb" in ballE)
-prefer 2
-apply (metis)
-apply (erule_tac x="ya" in ballE)
-prefer 2
-apply (metis)
-apply (metis in_respects)
-apply (erule conjE)+
-apply (erule bexE)
-apply rule
-apply (rule_tac x="xa" in bexI)
-apply metis
-apply metis
-apply rule+
-apply (erule_tac x="xb" in ballE)
-prefer 2
-apply (metis)
-apply (erule_tac x="ya" in ballE)
-prefer 2
-apply (metis)
-apply (metis in_respects)
-done
-
-lemma babs_rsp:
- assumes q: "Quotient R1 Abs1 Rep1"
- and a: "(R1 ===> R2) f g"
- shows "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
- apply (auto simp add: Babs_def)
- apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
- using a apply (simp add: Babs_def)
- apply (simp add: in_respects)
- using Quotient_rel[OF q]
- by metis
-
-lemma babs_prs:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- shows "(Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f)) \<equiv> f"
- apply(rule eq_reflection)
- apply(rule ext)
- apply simp
- apply (subgoal_tac "Rep1 x \<in> Respects R1")
- apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
- apply (simp add: in_respects Quotient_rel_rep[OF q1])
- done
-
-lemma babs_simp:
- assumes q: "Quotient R1 Abs Rep"
- shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
- apply(rule iffI)
- apply(simp_all only: babs_rsp[OF q])
- apply(auto simp add: Babs_def)
- apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
- apply(metis Babs_def)
- apply (simp add: in_respects)
- using Quotient_rel[OF q]
- by metis
-
-(* If a user proves that a particular functional relation
- is an equivalence this may be useful in regularising *)
-lemma babs_reg_eqv:
- shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
- by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)
-
-(* 3 lemmas needed for cleaning of quantifiers *)
-lemma all_prs:
- assumes a: "Quotient R absf repf"
- shows "Ball (Respects R) ((absf ---> id) f) = All f"
- using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
-by metis
-
-lemma ex_prs:
- assumes a: "Quotient R absf repf"
- shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
- using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
- by metis
-
-lemma ex1_prs:
- assumes a: "Quotient R absf repf"
- shows "((absf ---> id) ---> id) (Bexeq R) f = Ex1 f"
-apply simp
-apply (subst Bexeq_def)
-apply (subst Bex_def)
-apply (subst Ex1_def)
-apply simp
-apply rule
- apply (erule conjE)+
- apply (erule_tac exE)
- apply (erule conjE)
- apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
- apply (rule_tac x="absf x" in exI)
- apply (thin_tac "\<forall>x\<in>Respects R. \<forall>y\<in>Respects R. f (absf x) \<and> f (absf y) \<longrightarrow> R x y")
- apply (simp)
- apply rule+
- using a unfolding Quotient_def
- apply metis
- apply rule+
- apply (erule_tac x="x" in ballE)
- apply (erule_tac x="y" in ballE)
- apply simp
- apply (simp add: in_respects)
- apply (simp add: in_respects)
-apply (erule_tac exE)
- apply rule
- apply (rule_tac x="repf x" in exI)
- apply (simp only: in_respects)
- apply rule
- apply (metis Quotient_rel_rep[OF a])
-using a unfolding Quotient_def apply (simp)
-apply rule+
-using a unfolding Quotient_def in_respects
-apply metis
-done
-
-lemma fun_rel_id:
- assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
- shows "(R1 ===> R2) f g"
- using a by simp
-
-lemma fun_rel_id_asm:
- assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"
- shows "A \<longrightarrow> (R1 ===> R2) f g"
- using a by auto
-
-lemma quot_rel_rsp:
- assumes a: "Quotient R Abs Rep"
- shows "(R ===> R ===> op =) R R"
- apply(rule fun_rel_id)+
- apply(rule equals_rsp[OF a])
- apply(assumption)+
- done
-
-lemma o_prs:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- and q3: "Quotient R3 Abs3 Rep3"
- shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g"
- using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
- unfolding o_def expand_fun_eq by simp
-
-lemma o_rsp:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- and q3: "Quotient R3 Abs3 Rep3"
- and a1: "(R2 ===> R3) f1 f2"
- and a2: "(R1 ===> R2) g1 g2"
- shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
- using a1 a2 unfolding o_def expand_fun_eq
- by (auto)
-
-lemma cond_prs:
- assumes a: "Quotient R absf repf"
- shows "absf (if a then repf b else repf c) = (if a then b else c)"
- using a unfolding Quotient_def by auto
-
-lemma if_prs:
- assumes q: "Quotient R Abs Rep"
- shows "Abs (If a (Rep b) (Rep c)) = If a b c"
-using Quotient_abs_rep[OF q] by auto
-
-(* q not used *)
-lemma if_rsp:
- assumes q: "Quotient R Abs Rep"
- and a: "a1 = a2" "R b1 b2" "R c1 c2"
- shows "R (If a1 b1 c1) (If a2 b2 c2)"
-using a by auto
-
-lemma let_prs:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f"
- using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto
-
-lemma let_rsp:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and a1: "(R1 ===> R2) f g"
- and a2: "R1 x y"
- shows "R2 ((Let x f)::'c) ((Let y g)::'c)"
- using apply_rsp[OF q1 a1] a2 by auto
-
-
-
-
-(******************************************)
-(* REST OF THE FILE IS UNUSED (until now) *)
-(******************************************)
-
-lemma in_fun:
- shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"
- by (simp add: mem_def)
-
-lemma respects_thm:
- shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"
- unfolding Respects_def
- by (simp add: expand_fun_eq)
-
-lemma respects_rep_abs:
- assumes a: "Quotient R1 Abs1 Rep1"
- and b: "Respects (R1 ===> R2) f"
- and c: "R1 x x"
- shows "R2 (f (Rep1 (Abs1 x))) (f x)"
- using a b[simplified respects_thm] c unfolding Quotient_def
- by blast
-
-lemma respects_mp:
- assumes a: "Respects (R1 ===> R2) f"
- and b: "R1 x y"
- shows "R2 (f x) (f y)"
- using a b unfolding Respects_def
- by simp
-
-lemma respects_o:
- assumes a: "Respects (R2 ===> R3) f"
- and b: "Respects (R1 ===> R2) g"
- shows "Respects (R1 ===> R3) (f o g)"
- using a b unfolding Respects_def
- by simp
-
-lemma abs_o_rep:
- assumes a: "Quotient r absf repf"
- shows "absf o repf = id"
- apply(rule ext)
- apply(simp add: Quotient_abs_rep[OF a])
- done
-
-lemma eq_comp_r: "op = OO R OO op = \<equiv> R"
- apply (rule eq_reflection)
- apply (rule ext)+
- apply auto
- done
-
-lemma fun_rel_eq_rel:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g)
- \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"
- using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq
- by blast
-
-lemma let_babs:
- "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam"
- by (simp add: Babs_def)
-
-lemma fun_rel_equals:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- and r1: "Respects (R1 ===> R2) f"
- and r2: "Respects (R1 ===> R2) g"
- shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
- apply(rule_tac iffI)
- apply(rule)+
- apply (rule apply_rsp'[of "R1" "R2"])
- apply(subst Quotient_rel[OF fun_quotient[OF q1 q2]])
- apply auto
- using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
- apply (metis let_rsp q1)
- apply (metis fun_rel_eq_rel let_rsp q1 q2 r2)
- using r1 unfolding Respects_def expand_fun_eq
- apply(simp (no_asm_use))
- apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1])
- done
-
-(* ask Peter: fun_rel_IMP used twice *)
-lemma fun_rel_IMP2:
- assumes q1: "Quotient R1 Abs1 Rep1"
- and q2: "Quotient R2 Abs2 Rep2"
- and r1: "Respects (R1 ===> R2) f"
- and r2: "Respects (R1 ===> R2) g"
- and a: "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"
- shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"
- using q1 q2 r1 r2 a
- by (simp add: fun_rel_equals)
-
-lemma lambda_rep_abs_rsp:
- assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"
- and r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"
- shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"
- using r1 r2 by auto
-
-(* ask peter what are literal_case *)
-(* literal_case_PRS *)
-(* literal_case_RSP *)
-(* Cez: !f x. literal_case f x = f x *)
-
-(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
-lemma id_prs:
- assumes q: "Quotient R Abs Rep"
- shows "Abs (id (Rep e)) = id e"
- using Quotient_abs_rep[OF q] by auto
-
-lemma id_rsp:
- assumes q: "Quotient R Abs Rep"
- and a: "R e1 e2"
- shows "R (id e1) (id e2)"
- using a by auto
-
-end
-
--- a/Quot/quotient_tacs.ML Mon Jan 25 17:53:08 2010 +0100
+++ b/Quot/quotient_tacs.ML Mon Jan 25 18:13:44 2010 +0100
@@ -526,7 +526,8 @@
fun mk_simps thms = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver
val ss1 = mk_simps (defs @ prs @ @{thms babs_prs all_prs ex_prs ex1_prs})
- val ss2 = mk_simps (@{thms Quotient_abs_rep Quotient_rel_rep ex1_prs} @ ids)
+ val ss2 = mk_simps (@{thms Quotient_abs_rep[THEN eq_reflection]
+ Quotient_rel_rep[THEN eq_reflection] ex1_prs} @ ids)
in
EVERY' [simp_tac ss1,
fun_map_tac lthy,