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theory QuotScript
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imports Main
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begin
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definition
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"EQUIV E \<equiv> \<forall>x y. E x y = (E x = E y)"
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definition
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"REFL E \<equiv> \<forall>x. E x x"
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definition
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"SYM E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
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definition
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"TRANS E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
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lemma EQUIV_REFL_SYM_TRANS:
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shows "EQUIV E = (REFL E \<and> SYM E \<and> TRANS E)"
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unfolding EQUIV_def REFL_def SYM_def TRANS_def expand_fun_eq
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by (blast)
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lemma EQUIV_REFL:
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shows "EQUIV E ==> REFL E"
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by (simp add: EQUIV_REFL_SYM_TRANS)
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definition
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"PART_EQUIV 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)))"
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lemma EQUIV_IMP_PART_EQUIV:
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assumes a: "EQUIV E"
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shows "PART_EQUIV E"
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using a unfolding EQUIV_def PART_EQUIV_def
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by auto
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definition
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"QUOTIENT E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and>
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(\<forall>a. E (Rep a) (Rep a)) \<and>
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(\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
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lemma QUOTIENT_ABS_REP:
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assumes a: "QUOTIENT E Abs Rep"
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shows "Abs (Rep a) = a"
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using a unfolding QUOTIENT_def
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by simp
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lemma QUOTIENT_REP_REFL:
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assumes a: "QUOTIENT E Abs Rep"
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shows "E (Rep a) (Rep a)"
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using a unfolding QUOTIENT_def
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by blast
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lemma QUOTIENT_REL:
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assumes a: "QUOTIENT E Abs Rep"
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shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
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using a unfolding QUOTIENT_def
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by blast
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lemma QUOTIENT_REL_ABS:
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assumes a: "QUOTIENT E Abs Rep"
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shows "E r s \<Longrightarrow> Abs r = Abs s"
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using a unfolding QUOTIENT_def
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by blast
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lemma QUOTIENT_REL_ABS_EQ:
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assumes a: "QUOTIENT E Abs Rep"
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shows "E r r \<Longrightarrow> E s s \<Longrightarrow> E r s = (Abs r = Abs s)"
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using a unfolding QUOTIENT_def
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by blast
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lemma QUOTIENT_REL_REP:
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assumes a: "QUOTIENT E Abs Rep"
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shows "E (Rep a) (Rep b) = (a = b)"
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using a unfolding QUOTIENT_def
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by metis
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lemma QUOTIENT_REP_ABS:
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assumes a: "QUOTIENT E Abs Rep"
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shows "E r r \<Longrightarrow> E (Rep (Abs r)) r"
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using a unfolding QUOTIENT_def
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by blast
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lemma IDENTITY_EQUIV:
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shows "EQUIV (op =)"
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unfolding EQUIV_def
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by auto
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lemma IDENTITY_QUOTIENT:
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shows "QUOTIENT (op =) id id"
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unfolding QUOTIENT_def id_def
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by blast
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lemma QUOTIENT_SYM:
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assumes a: "QUOTIENT E Abs Rep"
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shows "SYM E"
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using a unfolding QUOTIENT_def SYM_def
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by metis
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lemma QUOTIENT_TRANS:
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assumes a: "QUOTIENT E Abs Rep"
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shows "TRANS E"
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using a unfolding QUOTIENT_def TRANS_def
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by metis
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fun
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prod_rel
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where
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"prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"
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fun
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fun_map
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where
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"fun_map f g h x = g (h (f x))"
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abbreviation
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fun_map_syn (infixr "--->" 55)
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where
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"f ---> g \<equiv> fun_map f g"
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lemma FUN_MAP_I:
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shows "(id ---> id) = id"
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by (simp add: expand_fun_eq id_def)
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lemma IN_FUN:
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shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"
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by (simp add: mem_def)
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fun
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FUN_REL
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where
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"FUN_REL E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
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abbreviation
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FUN_REL_syn (infixr "===>" 55)
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where
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"E1 ===> E2 \<equiv> FUN_REL E1 E2"
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lemma FUN_REL_EQ:
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"(op =) ===> (op =) = (op =)"
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by (simp add: expand_fun_eq)
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lemma FUN_QUOTIENT:
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assumes q1: "QUOTIENT R1 abs1 rep1"
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and q2: "QUOTIENT R2 abs2 rep2"
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shows "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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proof -
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have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
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apply(simp add: expand_fun_eq)
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using q1 q2
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apply(simp add: QUOTIENT_def)
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done
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moreover
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have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
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apply(auto)
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using q1 q2 unfolding QUOTIENT_def
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apply(metis)
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done
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moreover
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have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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(rep1 ---> abs2) r = (rep1 ---> abs2) s)"
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apply(auto simp add: expand_fun_eq)
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using q1 q2 unfolding QUOTIENT_def
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apply(metis)
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using q1 q2 unfolding QUOTIENT_def
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apply(metis)
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using q1 q2 unfolding QUOTIENT_def
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apply(metis)
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using q1 q2 unfolding QUOTIENT_def
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apply(metis)
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done
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ultimately
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show "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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unfolding QUOTIENT_def by blast
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qed
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definition
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Respects
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where
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"Respects R x \<equiv> (R x x)"
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lemma IN_RESPECTS:
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shows "(x \<in> Respects R) = R x x"
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unfolding mem_def Respects_def by simp
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lemma RESPECTS_THM:
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shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"
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unfolding Respects_def
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by (simp add: expand_fun_eq)
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lemma RESPECTS_MP:
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assumes a: "Respects (R1 ===> R2) f"
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and b: "R1 x y"
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shows "R2 (f x) (f y)"
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using a b unfolding Respects_def
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by simp
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lemma RESPECTS_REP_ABS:
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assumes a: "QUOTIENT R1 Abs1 Rep1"
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and b: "Respects (R1 ===> R2) f"
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and c: "R1 x x"
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shows "R2 (f (Rep1 (Abs1 x))) (f x)"
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using a b[simplified RESPECTS_THM] c unfolding QUOTIENT_def
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by blast
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lemma RESPECTS_o:
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assumes a: "Respects (R2 ===> R3) f"
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and b: "Respects (R1 ===> R2) g"
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shows "Respects (R1 ===> R3) (f o g)"
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using a b unfolding Respects_def
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by simp
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(*
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definition
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"RES_EXISTS_EQUIV R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and>
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(\<forall>x\<in> Respects R. \<forall>y\<in> Respects R. P x \<and> P y \<longrightarrow> R x y)"
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*)
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lemma FUN_REL_EQ_REL:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g)
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\<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"
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using FUN_QUOTIENT[OF q1 q2] unfolding Respects_def QUOTIENT_def expand_fun_eq
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by blast
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(* q1 and q2 not used; see next lemma *)
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lemma FUN_REL_MP:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
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by simp
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lemma FUN_REL_IMP:
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shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
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by simp
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lemma FUN_REL_EQUALS:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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and r1: "Respects (R1 ===> R2) f"
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and r2: "Respects (R1 ===> R2) g"
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shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
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apply(rule_tac iffI)
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using FUN_QUOTIENT[OF q1 q2] r1 r2 unfolding QUOTIENT_def Respects_def
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apply(metis FUN_REL_IMP)
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using r1 unfolding Respects_def expand_fun_eq
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apply(simp (no_asm_use))
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apply(metis QUOTIENT_REL[OF q2] QUOTIENT_REL_REP[OF q1])
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done
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(* ask Peter: FUN_REL_IMP used twice *)
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lemma FUN_REL_IMP2:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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and r1: "Respects (R1 ===> R2) f"
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and r2: "Respects (R1 ===> R2) g"
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and a: "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"
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shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"
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using q1 q2 r1 r2 a
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by (simp add: FUN_REL_EQUALS)
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lemma EQUALS_PRS:
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assumes q: "QUOTIENT R Abs Rep"
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shows "(x = y) = R (Rep x) (Rep y)"
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by (simp add: QUOTIENT_REL_REP[OF q])
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lemma EQUALS_RSP:
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assumes q: "QUOTIENT R Abs Rep"
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and a: "R x1 x2" "R y1 y2"
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shows "R x1 y1 = R x2 y2"
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using QUOTIENT_SYM[OF q] QUOTIENT_TRANS[OF q] unfolding SYM_def TRANS_def
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using a by blast
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lemma LAMBDA_PRS:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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253
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
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unfolding expand_fun_eq
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using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
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by simp
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lemma LAMBDA_PRS1:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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shows "(\<lambda>x. f x) = (Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x)"
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unfolding expand_fun_eq
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253
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
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e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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288 |
by (simp)
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e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
289 |
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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290 |
lemma APP_PRS:
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e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
291 |
assumes q1: "QUOTIENT R1 abs1 rep1"
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
292 |
and q2: "QUOTIENT R2 abs2 rep2"
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
293 |
shows "abs2 ((abs1 ---> rep2) f (rep1 x)) = f x"
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
294 |
unfolding expand_fun_eq
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
295 |
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
|
e169a99c6ada
Automatic computation of application preservation and manually finished "alpha.induct". Slow...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
296 |
by simp
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(* Ask Peter: assumption q1 and q2 not used and lemma is the 'identity' *)
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lemma LAMBDA_RSP:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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and a: "(R1 ===> R2) f1 f2"
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shows "(R1 ===> R2) (\<lambda>x. f1 x) (\<lambda>y. f2 y)"
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by (rule a)
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(* ASK Peter about next four lemmas in quotientScript
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lemma ABSTRACT_PRS:
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assumes q1: "QUOTIENT R1 Abs1 Rep1"
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and q2: "QUOTIENT R2 Abs2 Rep2"
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shows "f = (Rep1 ---> Abs2) ???"
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*)
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lemma LAMBDA_REP_ABS_RSP:
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assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"
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and r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"
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shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"
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using r1 r2 by auto
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lemma REP_ABS_RSP:
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assumes q: "QUOTIENT R Abs Rep"
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and a: "R x1 x2"
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shows "R x1 (Rep (Abs x2))"
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and "R (Rep (Abs x1)) x2"
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proof -
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show "R x1 (Rep (Abs x2))"
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using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q])
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next
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show "R (Rep (Abs x1)) x2"
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using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q] QUOTIENT_SYM[of q])
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qed
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0
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(* ----------------------------------------------------- *)
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(* Quantifiers: FORALL, EXISTS, EXISTS_UNIQUE, *)
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|
334 |
(* RES_FORALL, RES_EXISTS, RES_EXISTS_EQUIV *)
|
|
335 |
(* ----------------------------------------------------- *)
|
|
336 |
|
|
337 |
(* what is RES_FORALL *)
|
|
338 |
(*--`!R (abs:'a -> 'b) rep. QUOTIENT R abs rep ==>
|
|
339 |
!f. $! f = RES_FORALL (respects R) ((abs --> I) f)`--*)
|
|
340 |
(*as peter here *)
|
|
341 |
|
|
342 |
(* bool theory: COND, LET *)
|
|
343 |
|
|
344 |
lemma IF_PRS:
|
|
345 |
assumes q: "QUOTIENT R Abs Rep"
|
|
346 |
shows "If a b c = Abs (If a (Rep b) (Rep c))"
|
|
347 |
using QUOTIENT_ABS_REP[OF q] by auto
|
|
348 |
|
|
349 |
(* ask peter: no use of q *)
|
|
350 |
lemma IF_RSP:
|
|
351 |
assumes q: "QUOTIENT R Abs Rep"
|
|
352 |
and a: "a1 = a2" "R b1 b2" "R c1 c2"
|
|
353 |
shows "R (If a1 b1 c1) (If a2 b2 c2)"
|
|
354 |
using a by auto
|
|
355 |
|
|
356 |
lemma LET_PRS:
|
|
357 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
358 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
359 |
shows "Let x f = Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f))"
|
|
360 |
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto
|
|
361 |
|
|
362 |
lemma LET_RSP:
|
|
363 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
364 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
365 |
and a1: "(R1 ===> R2) f g"
|
|
366 |
and a2: "R1 x y"
|
|
367 |
shows "R2 (Let x f) (Let y g)"
|
|
368 |
using FUN_REL_MP[OF q1 q2 a1] a2
|
|
369 |
by auto
|
|
370 |
|
|
371 |
|
|
372 |
(* ask peter what are literal_case *)
|
|
373 |
(* literal_case_PRS *)
|
|
374 |
(* literal_case_RSP *)
|
|
375 |
|
|
376 |
|
|
377 |
(* FUNCTION APPLICATION *)
|
|
378 |
|
|
379 |
lemma APPLY_PRS:
|
|
380 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
381 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
382 |
shows "f x = Abs2 (((Abs1 ---> Rep2) f) (Rep1 x))"
|
|
383 |
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto
|
|
384 |
|
|
385 |
(* ask peter: no use of q1 q2 *)
|
|
386 |
lemma APPLY_RSP:
|
|
387 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
388 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
389 |
and a: "(R1 ===> R2) f g" "R1 x y"
|
|
390 |
shows "R2 (f x) (g y)"
|
|
391 |
using a by (rule FUN_REL_IMP)
|
|
392 |
|
317
|
393 |
lemma APPLY_RSP2:
|
|
394 |
assumes a: "(R1 ===> R2) f g" "R1 x y"
|
|
395 |
shows "R2 (f x) (g y)"
|
|
396 |
using a by (rule FUN_REL_IMP)
|
|
397 |
|
0
|
398 |
|
|
399 |
(* combinators: I, K, o, C, W *)
|
|
400 |
|
|
401 |
lemma I_PRS:
|
|
402 |
assumes q: "QUOTIENT R Abs Rep"
|
126
|
403 |
shows "id e = Abs (id (Rep e))"
|
0
|
404 |
using QUOTIENT_ABS_REP[OF q] by auto
|
|
405 |
|
|
406 |
lemma I_RSP:
|
|
407 |
assumes q: "QUOTIENT R Abs Rep"
|
|
408 |
and a: "R e1 e2"
|
126
|
409 |
shows "R (id e1) (id e2)"
|
0
|
410 |
using a by auto
|
|
411 |
|
|
412 |
lemma o_PRS:
|
|
413 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
414 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
415 |
and q3: "QUOTIENT R3 Abs3 Rep3"
|
|
416 |
shows "f o g = (Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g))"
|
|
417 |
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] QUOTIENT_ABS_REP[OF q3]
|
|
418 |
unfolding o_def expand_fun_eq
|
|
419 |
by simp
|
|
420 |
|
|
421 |
lemma o_RSP:
|
|
422 |
assumes q1: "QUOTIENT R1 Abs1 Rep1"
|
|
423 |
and q2: "QUOTIENT R2 Abs2 Rep2"
|
|
424 |
and q3: "QUOTIENT R3 Abs3 Rep3"
|
|
425 |
and a1: "(R2 ===> R3) f1 f2"
|
|
426 |
and a2: "(R1 ===> R2) g1 g2"
|
|
427 |
shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
|
|
428 |
using a1 a2 unfolding o_def expand_fun_eq
|
|
429 |
by (auto)
|
|
430 |
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
431 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
432 |
(* TODO: Put the following lemmas in proper places *)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
433 |
|
93
|
434 |
lemma equiv_res_forall:
|
|
435 |
fixes P :: "'a \<Rightarrow> bool"
|
|
436 |
assumes a: "EQUIV E"
|
|
437 |
shows "Ball (Respects E) P = (All P)"
|
|
438 |
using a by (metis EQUIV_def IN_RESPECTS a)
|
|
439 |
|
|
440 |
lemma equiv_res_exists:
|
|
441 |
fixes P :: "'a \<Rightarrow> bool"
|
|
442 |
assumes a: "EQUIV E"
|
|
443 |
shows "Bex (Respects E) P = (Ex P)"
|
|
444 |
using a by (metis EQUIV_def IN_RESPECTS a)
|
|
445 |
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
446 |
lemma FORALL_REGULAR:
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
447 |
assumes a: "!x :: 'a. (P x --> Q x)"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
448 |
and b: "All P"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
449 |
shows "All Q"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
450 |
using a b by (metis)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
451 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
452 |
lemma EXISTS_REGULAR:
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
453 |
assumes a: "!x :: 'a. (P x --> Q x)"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
454 |
and b: "Ex P"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
455 |
shows "Ex Q"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
456 |
using a b by (metis)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
457 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
458 |
lemma RES_FORALL_REGULAR:
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
459 |
assumes a: "!x :: 'a. (R x --> P x --> Q x)"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
460 |
and b: "Ball R P"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
461 |
shows "Ball R Q"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
462 |
using a b by (metis COMBC_def Collect_def Collect_mem_eq)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
463 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
464 |
lemma RES_EXISTS_REGULAR:
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
465 |
assumes a: "!x :: 'a. (R x --> P x --> Q x)"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
466 |
and b: "Bex R P"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
467 |
shows "Bex R Q"
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
468 |
using a b by (metis COMBC_def Collect_def Collect_mem_eq)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
469 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
470 |
lemma LEFT_RES_FORALL_REGULAR:
|
396
|
471 |
assumes a: "\<And>x. (R x \<and> (Q x \<longrightarrow> P x))"
|
|
472 |
shows "Ball R Q \<longrightarrow> All P"
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
473 |
using a
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
474 |
apply (metis COMBC_def Collect_def Collect_mem_eq a)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
475 |
done
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
476 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
477 |
lemma RIGHT_RES_FORALL_REGULAR:
|
396
|
478 |
assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
|
252
e30997c88050
Regularize for equalities and a better tactic. "alpha.cases" now lifts.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
479 |
shows "All P \<longrightarrow> Ball R Q"
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
480 |
using a
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
481 |
apply (metis COMBC_def Collect_def Collect_mem_eq a)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
482 |
done
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
483 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
484 |
lemma LEFT_RES_EXISTS_REGULAR:
|
396
|
485 |
assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
|
395
|
486 |
shows "Bex R Q \<longrightarrow> Ex P"
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
487 |
using a by (metis COMBC_def Collect_def Collect_mem_eq)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
488 |
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
489 |
lemma RIGHT_RES_EXISTS_REGULAR:
|
396
|
490 |
assumes a: "\<And>x. (R x \<and> (P x \<longrightarrow> Q x))"
|
395
|
491 |
shows "Ex P \<longrightarrow> Bex R Q"
|
96
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
492 |
using a by (metis COMBC_def Collect_def Collect_mem_eq)
|
4da714704611
A number of lemmas for REGULARIZE_TAC and regularizing card1.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
493 |
|
162
|
494 |
(* TODO: Add similar *)
|
153
|
495 |
lemma RES_FORALL_RSP:
|
|
496 |
shows "\<And>f g. (R ===> (op =)) f g ==>
|
|
497 |
(Ball (Respects R) f = Ball (Respects R) g)"
|
155
|
498 |
apply (simp add: FUN_REL.simps Ball_def IN_RESPECTS)
|
|
499 |
done
|
153
|
500 |
|
171
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
501 |
lemma RES_EXISTS_RSP:
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
502 |
shows "\<And>f g. (R ===> (op =)) f g ==>
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
503 |
(Bex (Respects R) f = Bex (Respects R) g)"
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
504 |
apply (simp add: FUN_REL.simps Bex_def IN_RESPECTS)
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
505 |
done
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
506 |
|
188
|
507 |
|
162
|
508 |
lemma FORALL_PRS:
|
|
509 |
assumes a: "QUOTIENT R absf repf"
|
183
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
510 |
shows "All f = Ball (Respects R) ((absf ---> id) f)"
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
511 |
using a
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
512 |
unfolding QUOTIENT_def
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
513 |
by (metis IN_RESPECTS fun_map.simps id_apply)
|
162
|
514 |
|
171
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
515 |
lemma EXISTS_PRS:
|
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
516 |
assumes a: "QUOTIENT R absf repf"
|
183
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
517 |
shows "Ex f = Bex (Respects R) ((absf ---> id) f)"
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
518 |
using a
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
519 |
unfolding QUOTIENT_def
|
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
520 |
by (metis COMBC_def Collect_def Collect_mem_eq IN_RESPECTS fun_map.simps id_apply mem_def)
|
187
|
521 |
|
|
522 |
lemma COND_PRS:
|
|
523 |
assumes a: "QUOTIENT R absf repf"
|
|
524 |
shows "(if a then b else c) = absf (if a then repf b else repf c)"
|
188
|
525 |
using a unfolding QUOTIENT_def by auto
|
187
|
526 |
|
|
527 |
(* These are the fixed versions, general ones need to be proved. *)
|
|
528 |
lemma ho_all_prs:
|
228
|
529 |
shows "((op = ===> op =) ===> op =) All All"
|
183
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
530 |
by auto
|
171
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
531 |
|
228
|
532 |
lemma ho_ex_prs:
|
|
533 |
shows "((op = ===> op =) ===> op =) Ex Ex"
|
183
6acf9e001038
proved the two lemmas in QuotScript (reformulated them without leading forall)
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
534 |
by auto
|
171
13aab4c59096
More infrastructure for automatic lifting of theorems lifted before
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
535 |
|
93
|
536 |
end
|
95
|
537 |
|