diff -r dd6ce36a0616 -r 243a5ceaa088 Quot/QuotBase.thy
--- a/Quot/QuotBase.thy	Wed Feb 10 21:39:40 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,626 +0,0 @@
-(*  Title:      QuotBase.thy
-    Author:     Cezary Kaliszyk and Christian Urban
-*)
-
-theory QuotBase
-imports Plain ATP_Linkup
-begin
-
-text {*
-  Basic definition for equivalence relations
-  that are represented by predicates.
-*}
-
-definition
-  "equivp E \<longleftrightarrow> (\<forall>x y. E x y = (E x = E y))"
-
-definition
-  "reflp E \<longleftrightarrow> (\<forall>x. E x x)"
-
-definition
-  "symp E \<longleftrightarrow> (\<forall>x y. E x y \<longrightarrow> E y x)"
-
-definition
-  "transp E \<longleftrightarrow> (\<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> E x x"
-  by (simp only: equivp_reflp_symp_transp reflp_def)
-
-lemma equivp_symp:
-  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
-  by (metis equivp_reflp_symp_transp symp_def)
-
-lemma equivp_transp:
-  shows "equivp E \<Longrightarrow> 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)
-
-lemma identity_equivp:
-  shows "equivp (op =)"
-  unfolding equivp_def
-  by auto
-
-
-text {* Partial equivalences: not yet used anywhere *}
-definition
-  "part_equivp E \<longleftrightarrow> ((\<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_implies_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"
-
-lemma eq_comp_r: 
-  shows "((op =) OOO R) = R"
-  by (auto simp add: expand_fun_eq)
-
-section {* Respects predicate *}
-
-definition
-  Respects
-where
-  "Respects R x \<longleftrightarrow> (R x x)"
-
-lemma in_respects:
-  shows "(x \<in> Respects R) = R x x"
-  unfolding mem_def Respects_def 
-  by simp
-
-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 =)) = (op =)"
-  by (simp add: expand_fun_eq)
-
-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
-
-
-section {* Quotient Predicate *}
-
-definition
-  "Quotient E Abs Rep \<longleftrightarrow>
-     (\<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
-
-lemma identity_quotient:
-  shows "Quotient (op =) id id"
-  unfolding Quotient_def id_def
-  by blast
-
-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"
-    using q1 q2 
-    unfolding Quotient_def
-    unfolding expand_fun_eq
-    by simp
-  moreover
-  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
-    using q1 q2 
-    unfolding Quotient_def
-    by (simp (no_asm)) (metis)
-  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)"
-    unfolding expand_fun_eq
-    apply(auto)
-    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
-
-lemma abs_o_rep:
-  assumes a: "Quotient R Abs Rep"
-  shows "Abs o Rep = id"
-  unfolding expand_fun_eq
-  by (simp add: Quotient_abs_rep[OF a])
-
-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 a Quotient_symp[OF q] Quotient_transp[OF q] 
-  unfolding symp_def transp_def
-  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 a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
-  by metis
-
-lemma rep_abs_rsp_left:
-  assumes q: "Quotient R Abs Rep"
-  and     a: "R x1 x2"
-  shows "R (Rep (Abs x1)) x2"
-  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
-  by metis
-
-text{* 
-  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
-
-section {* 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)"
-  using a 
-  unfolding equivp_def
-  by (auto simp add: in_respects)
-
-lemma bex_reg_eqv:
-  fixes P :: "'a \<Rightarrow> bool"
-  assumes a: "equivp R"
-  shows "Bex (Respects R) P = (Ex P)"
-  using a 
-  unfolding equivp_def
-  by (auto simp add: in_respects)
-
-lemma ball_reg_right:
-  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
-  shows "All P \<longrightarrow> Ball R Q"
-  using a by (metis COMBC_def Collect_def Collect_mem_eq)
-
-lemma bex_reg_left:
-  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
-  shows "Bex R Q \<longrightarrow> Ex P"
-  using a by (metis COMBC_def Collect_def Collect_mem_eq)
-
-lemma ball_reg_left:
-  assumes a: "equivp R"
-  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
-  using a by (metis equivp_reflp in_respects)
-
-lemma bex_reg_right:
-  assumes a: "equivp R"
-  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
-  using a by (metis equivp_reflp in_respects)
-
-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:
-  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
-  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
-  using assms by auto
-
-lemma bex_ex_comm:
-  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
-  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
-  using assms by auto
-
-section {* 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"
-
-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 in_respects)
-  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))) = f"
-  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 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 "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
-  using a 
-  by (simp add: Ex1_def in_respects) auto
-
-(* 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 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
-
-section {* Bex1_rel quantifier *}
-
-definition
-  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
-where
-  "Bex1_rel R P \<longleftrightarrow> (\<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)))"
-
-lemma bex1_rel_aux:
-  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
-  unfolding Bex1_rel_def
-  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="xaa" in ballE)
-  prefer 2
-  apply (metis)
-  apply (erule_tac x="ya" in ballE)
-  prefer 2
-  apply (metis)
-  apply (metis in_respects)
-  done
-
-lemma bex1_rel_aux2:
-  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
-  unfolding Bex1_rel_def
-  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="xaa" in ballE)
-  prefer 2
-  apply (metis)
-  apply (erule_tac x="ya" in ballE)
-  prefer 2
-  apply (metis)
-  apply (metis in_respects)
-  done
-
-lemma bex1_rel_rsp:
-  assumes a: "Quotient R absf repf"
-  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
-  apply simp
-  apply clarify
-  apply rule
-  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
-  apply (erule bex1_rel_aux2)
-  apply assumption
-  done
-
-
-lemma ex1_prs:
-  assumes a: "Quotient R absf repf"
-  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
-apply simp
-apply (subst Bex1_rel_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 (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 bex1_bexeq_reg: "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
-  apply (simp add: Ex1_def Bex1_rel_def in_respects)
-  apply clarify
-  apply auto
-  apply (rule bexI)
-  apply assumption
-  apply (simp add: in_respects)
-  apply (simp add: in_respects)
-  apply auto
-  done
-
-section {* Various respects and preserve lemmas *}
-
-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
-
-end
-