
diff -r fe468f8723fc -r d030c8e19465 QuotScript.thy
--- a/QuotScript.thy	Fri Dec 04 17:15:55 2009 +0100
+++ b/QuotScript.thy	Fri Dec 04 17:36:45 2009 +0100
@@ -2,13 +2,13 @@
 imports Plain ATP_Linkup
 begin
 
-definition 
-  "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)" 
+definition
+  "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
 
 definition
   "reflp E \<equiv> \<forall>x. E x x"
 
-definition 
+definition
   "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
 
 definition
@@ -16,8 +16,8 @@
 
 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)
+  unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq
+  by (blast)
 
 lemma equivp_refl:
   shows "equivp R \<Longrightarrow> (\<And>x. R x x)"
@@ -33,66 +33,66 @@
 lemma equivp_IMP_part_equivp:
   assumes a: "equivp E"
   shows "part_equivp E"
-using a unfolding equivp_def part_equivp_def
-by auto
+  using a unfolding equivp_def part_equivp_def
+  by auto
 
 definition
-  "Quotient E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and> 
-                        (\<forall>a. E (Rep a) (Rep a)) \<and> 
+  "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
+  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
+  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
+  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
+  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
+  using a unfolding Quotient_def
+  by blast
 
 lemma identity_equivp:
   shows "equivp (op =)"
-unfolding equivp_def
-by auto
+  unfolding equivp_def
+  by auto
 
 lemma identity_quotient:
   shows "Quotient (op =) id id"
-unfolding Quotient_def id_def
-by blast
+  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
+  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
+  using a unfolding Quotient_def transp_def
+  by metis
 
 fun
   prod_rel
@@ -104,7 +104,6 @@
 where
   "fun_map f g h x = g (h (f x))"
 
-
 abbreviation
   fun_map_syn (infixr "--->" 55)
 where
@@ -112,7 +111,7 @@
 
 lemma fun_map_id:
   shows "(id ---> id) = id"
-by (simp add: expand_fun_eq id_def)
+  by (simp add: expand_fun_eq id_def)
 
 fun
   fun_rel
@@ -169,155 +168,55 @@
 
 lemma in_respects:
   shows "(x \<in> Respects R) = R x x"
-unfolding mem_def Respects_def by simp
-
-(* TODO: it is the same as APPLY_RSP *)
-(* q1 and q2 not used; see next lemma *)
-lemma fun_rel_MP:
-  assumes q1: "Quotient R1 Abs1 Rep1"
-  and     q2: "Quotient R2 Abs2 Rep2"
-  shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
-by simp
-
-lemma fun_rel_IMP:
-  shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
-by simp
-
+  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
+  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
+  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
+  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])
-
-(* ----------------------------------------------------- *)
-(* Quantifiers: FORALL, EXISTS, EXISTS_UNIQUE,           *)
-(*              Ball, Bex, RES_EXISTS_EQUIV              *)
-(* ----------------------------------------------------- *)
-
-(* bool theory: COND, LET *)
-
-lemma IF_PRS:
-  assumes q: "Quotient R Abs Rep"
-  shows "If a b c = Abs (If a (Rep b) (Rep c))"
-using Quotient_abs_rep[OF q] by auto
-
-(* ask peter: no use of q *)
-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 "Let x f = Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f))"
-using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto
-
-lemma LET_RSP:
-  assumes q1: "Quotient R1 Abs1 Rep1"
-  and     q2: "Quotient R2 Abs2 Rep2"
-  and     a1: "(R1 ===> R2) f g"
-  and     a2: "R1 x y"
-  shows "R2 (Let x f) (Let y g)"
-using fun_rel_MP[OF q1 q2 a1] a2
-by auto
-
-
-(* ask peter what are literal_case *)
-(* literal_case_PRS *)
-(* literal_case_RSP *)
-
-
-(* FUNCTION APPLICATION *)
+  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 R2 that
-   will be provable; which is why we need to use APPLY_RSP *)
+   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:
   assumes q: "Quotient R1 Abs1 Rep1"
   and     a: "(R1 ===> R2) f g" "R1 x y"
   shows "R2 ((f::'a\<Rightarrow>'c) x) ((g::'a\<Rightarrow>'c) y)"
-using a by (rule fun_rel_IMP)
+  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 (rule fun_rel_IMP)
-
-
-(* combinators: I, K, o, C, W *)
-
-(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
-lemma I_PRS:
-  assumes q: "Quotient R Abs Rep"
-  shows "id e = Abs (id (Rep e))"
-using Quotient_abs_rep[OF q] by auto
-
-lemma I_RSP:
-  assumes q: "Quotient R Abs Rep"
-  and     a: "R e1 e2"
-  shows "R (id e1) (id e2)"
-using a by auto
-
-lemma o_PRS:
-  assumes q1: "Quotient R1 Abs1 Rep1"
-  and     q2: "Quotient R2 Abs2 Rep2"
-  and     q3: "Quotient R3 Abs3 Rep3"
-  shows "f o g = (Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) 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 "(if a then b else c) = absf (if a then repf b else repf c)"
-  using a unfolding Quotient_def by auto
-
-
-
-
+  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"
@@ -437,7 +336,13 @@
   by (metis COMBC_def Collect_def Collect_mem_eq in_respects fun_map.simps id_apply)
 
 
-(* UNUSED *)
+
+
+
+
+(******************************************)
+(* REST OF THE FILE IS UNUSED (until now) *)
+(******************************************)
 lemma Quotient_rel_abs:
   assumes a: "Quotient E Abs Rep"
   shows "E r s \<Longrightarrow> Abs r = Abs s"
@@ -522,7 +427,7 @@
   shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
 apply(rule_tac iffI)
 using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
-apply(metis fun_rel_IMP)
+apply(metis apply_rsp')
 using r1 unfolding Respects_def expand_fun_eq
 apply(simp (no_asm_use))
 apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1])
@@ -553,5 +458,83 @@
 using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
 
 
+
+(* bool theory: COND, LET *)
+lemma IF_PRS:
+  assumes q: "Quotient R Abs Rep"
+  shows "If a b c = Abs (If a (Rep b) (Rep c))"
+using Quotient_abs_rep[OF q] by auto
+
+(* ask peter: no use of q *)
+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 "Let x f = Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) 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
+
+
+
+(* ask peter what are literal_case *)
+(* literal_case_PRS *)
+(* literal_case_RSP *)
+
+
+
+
+
+(* combinators: I, K, o, C, W *)
+
+(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
+
+lemma I_PRS:
+  assumes q: "Quotient R Abs Rep"
+  shows "id e = Abs (id (Rep e))"
+using Quotient_abs_rep[OF q] by auto
+
+lemma I_RSP:
+  assumes q: "Quotient R Abs Rep"
+  and     a: "R e1 e2"
+  shows "R (id e1) (id e2)"
+using a by auto
+
+lemma o_PRS:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  and     q3: "Quotient R3 Abs3 Rep3"
+  shows "f o g = (Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) 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 "(if a then b else c) = absf (if a then repf b else repf c)"
+  using a unfolding Quotient_def by auto
+
+
 end
 




