diff -r e68f501f76d0 -r 19032e71fb1c Quot/QuotScript.thy --- a/Quot/QuotScript.thy Fri Dec 11 17:22:26 2009 +0100 +++ b/Quot/QuotScript.thy Fri Dec 11 17:59:29 2009 +0100 @@ -29,12 +29,12 @@ lemma equivp_transp: shows "equivp E \ (\x y z. E x y \ E y z \ E x z)" -by (metis equivp_reflp_symp_transp transp_def) + 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) + using assms by (simp add: equivp_reflp_symp_transp) definition "part_equivp E \ (\x. E x x) \ (\x y. E x y = (E x x \ E y y \ (E x = E y)))" @@ -81,6 +81,12 @@ using a unfolding Quotient_def by blast +lemma Quotient_rel_abs: + assumes a: "Quotient E Abs Rep" + shows "E r s \ Abs r = Abs s" + using a unfolding Quotient_def + by blast + lemma identity_equivp: shows "equivp (op =)" unfolding equivp_def @@ -129,7 +135,7 @@ lemma fun_rel_eq: "(op =) ===> (op =) \ (op =)" -by (rule eq_reflection) (simp add: expand_fun_eq) + by (rule eq_reflection) (simp add: expand_fun_eq) lemma fun_quotient: assumes q1: "Quotient R1 abs1 rep1" @@ -203,6 +209,12 @@ 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 @@ -309,11 +321,11 @@ lemma ball_all_comm: "(\y. (\x\P. A x y) \ (\x. B x y)) \ ((\x\P. \y. A x y) \ (\x. \y. B x y))" -by auto + by auto lemma bex_ex_comm: "((\y. \x. A x y) \ (\y. \x\P. B x y)) \ ((\x. \y. A x y) \ (\x\P. \y. B x y))" -by auto + by auto (* Bounded abstraction *) definition @@ -358,20 +370,20 @@ 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 \ Respects R1 \ y \ Respects R1") -apply(metis Babs_def) -apply (simp add: in_respects) -using Quotient_rel[OF q] -by metis + apply(rule iffI) + apply(simp_all only: babs_rsp[OF q]) + apply(auto simp add: Babs_def) + apply (subgoal_tac "x \ Respects R1 \ y \ Respects R1") + apply(metis Babs_def) + apply (simp add: in_respects) + using Quotient_rel[OF q] + by metis - -(* needed for regularising? *) +(* 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 \ Babs (Respects R) P = P" -by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp) + by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp) (* 2 lemmas needed for cleaning of quantifiers *) lemma all_prs: @@ -389,12 +401,12 @@ lemma fun_rel_id: assumes a: "\x y. R1 x y \ R2 (f x) (g y)" shows "(R1 ===> R2) f g" -using a by simp + using a by simp lemma fun_rel_id_asm: assumes a: "\x y. R1 x y \ (A \ R2 (f x) (g y))" shows "A \ (R1 ===> R2) f g" -using a by auto + using a by auto lemma quot_rel_rsp: assumes a: "Quotient R Abs Rep" @@ -404,7 +416,53 @@ 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 @@ -413,104 +471,62 @@ (* REST OF THE FILE IS UNUSED (until now) *) (******************************************) -(* Always safe to apply, but not too helpful *) -lemma app_prs2: - assumes q1: "Quotient R1 abs1 rep1" - and q2: "Quotient R2 abs2 rep2" - shows "((abs1 ---> rep2) ((rep1 ---> abs2) f) (rep1 x)) = rep2 (((rep1 ---> abs2) f) x)" -unfolding expand_fun_eq -using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] -by simp - -lemma Quotient_rel_abs: - assumes a: "Quotient E Abs Rep" - shows "E r s \ 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 \ E s s \ E r s = (Abs r = Abs s)" -using a unfolding Quotient_def -by blast - lemma in_fun: shows "x \ ((f ---> g) s) = g (f x \ s)" -by (simp add: mem_def) + by (simp add: mem_def) -lemma RESPECTS_THM: +lemma respects_thm: shows "Respects (R1 ===> R2) f = (\x y. R1 x y \ R2 (f x) (f y))" -unfolding Respects_def -by (simp add: expand_fun_eq) + unfolding Respects_def + by (simp add: expand_fun_eq) -lemma RESPECTS_REP_ABS: +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 + using a b[simplified respects_thm] c unfolding Quotient_def + by blast -lemma RESPECTS_MP: +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 + using a b unfolding Respects_def + by simp -lemma RESPECTS_o: +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 + using a b unfolding Respects_def + by simp -lemma fun_rel_EQ_REL: +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) \ (Respects (R1 ===> R2) g) + shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \ (Respects (R1 ===> R2) g) \ ((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 + using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq + by blast -(* 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) (\x. f1 x) (\y. f2 y)" -by (rule a) +lemma let_babs: + "v \ r \ Let v (Babs r lam) = Let v lam" + by (simp add: Babs_def) -(* 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: +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) = (\x y. R1 x y \ 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 + 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: @@ -520,100 +536,31 @@ and r2: "Respects (R1 ===> R2) g" and a: "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g" shows "R1 x y \ R2 (f x) (g y)" -using q1 q2 r1 r2 a -by (simp add: fun_rel_EQUALS) + using q1 q2 r1 r2 a + by (simp add: fun_rel_equals) -lemma LAMBDA_REP_ABS_RSP: +lemma lambda_rep_abs_rsp: assumes r1: "\r r'. R1 r r' \R1 r (Rep1 (Abs1 r'))" and r2: "\r r'. R2 r r' \R2 r (Rep2 (Abs2 r'))" shows "(R1 ===> R2) f1 f2 \ (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 (Rep (Abs x1)) 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 - - + using r1 r2 by auto (* ask peter what are literal_case *) (* literal_case_PRS *) (* literal_case_RSP *) - - - - - -(* combinators: I, K, o, C, W *) +(* 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 I_PRS: +lemma id_prs: assumes q: "Quotient R Abs Rep" - shows "id e = Abs (id (Rep e))" -using Quotient_abs_rep[OF q] by auto + shows "Abs (id (Rep e)) = id e" + using Quotient_abs_rep[OF q] by auto -lemma I_RSP: +lemma id_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 auto end