diff -r 6088fea1c8b1 -r 8a1c8dc72b5c Quot/QuotScript.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/QuotScript.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,569 @@
+theory QuotScript
+imports Plain ATP_Linkup
+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)
+
+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
+
+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) \<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 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
+
+fun
+  fun_map
+where
+  "fun_map f g h x = g (h (f x))"
+
+abbreviation
+  fun_map_syn (infixr "--->" 55)
+where
+  "f ---> g \<equiv> fun_map f g"
+
+lemma fun_map_id:
+  shows "(id ---> id) = id"
+  by (simp add: expand_fun_eq id_def)
+
+fun
+  fun_rel
+where
+  "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
+
+abbreviation
+  fun_rel_syn (infixr "===>" 55)
+where
+  "E1 ===> E2 \<equiv> fun_rel E1 E2"
+
+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])
+
+(* 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:
+  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 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: Respects_def 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:
+  fixes P::"'a \<Rightarrow> bool"
+  and x::"'a"
+  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)"
+
+(* 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 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
+
+(* 2 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
+  by (metis in_respects fun_map.simps id_apply)
+
+lemma ex_prs:
+  assumes a: "Quotient R absf repf"
+  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
+  using a unfolding Quotient_def
+  by (metis COMBC_def Collect_def Collect_mem_eq in_respects fun_map.simps id_apply)
+
+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 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
+
+
+
+
+
+
+(******************************************)
+(* 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"
+using a unfolding Quotient_def
+by blast
+
+lemma Quotient_rel_abs_eq:
+  assumes a: "Quotient E Abs Rep"
+  shows "E r r \<Longrightarrow> E s s \<Longrightarrow> E r s = (Abs r = Abs s)"
+using a unfolding Quotient_def
+by blast
+
+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 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
+
+(* Not used since in the end we just unfold fun_map *)
+lemma APP_PRS:
+  assumes q1: "Quotient R1 abs1 rep1"
+  and     q2: "Quotient R2 abs2 rep2"
+  shows "abs2 ((abs1 ---> rep2) f (rep1 x)) = f x"
+unfolding expand_fun_eq
+using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
+by simp
+
+(* Ask Peter: assumption q1 and q2 not used and lemma is the 'identity' *)
+lemma LAMBDA_RSP:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  and     a: "(R1 ===> R2) f1 f2"
+  shows "(R1 ===> R2) (\<lambda>x. f1 x) (\<lambda>y. f2 y)"
+by (rule a)
+
+(* ASK Peter about next four lemmas in quotientScript
+lemma ABSTRACT_PRS:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  shows "f = (Rep1 ---> Abs2) ???"
+*)
+
+
+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)
+using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
+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])
+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
+
+(* Not used *)
+lemma rep_abs_rsp_left:
+  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])
+
+
+
+(* 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
+