theory Nominal2_FCB imports Nominal2_Abs begin text ‹ A tactic which solves all trivial cases in function definitions, and leaves the others unchanged. › ML ‹ val all_trivials : (Proof.context -> Proof.method) context_parser = Scan.succeed (fn ctxt => let val tac = TRYALL (SOLVED' (full_simp_tac ctxt)) in Method.SIMPLE_METHOD' (K tac) end) › method_setup all_trivials = ‹all_trivials› ‹solves trivial goals› lemma Abs_lst1_fcb: fixes x y :: "'a :: at" and S T :: "'b :: fs" assumes e: "[[atom x]]lst. T = [[atom y]]lst. S" and f1: "⟦x ≠ y; atom y ♯ T; atom x ♯ (y ↔ x) ∙ T⟧ ⟹ atom x ♯ f x T" and f2: "⟦x ≠ y; atom y ♯ T; atom x ♯ (y ↔ x) ∙ T⟧ ⟹ atom y ♯ f x T" and p: "⟦S = (x ↔ y) ∙ T; x ≠ y; atom y ♯ T; atom x ♯ S⟧ ⟹ (x ↔ y) ∙ (f x T) = f y S" shows "f x T = f y S" using e apply(case_tac "atom x ♯ S") apply(simp add: Abs1_eq_iff') apply(elim conjE disjE) apply(simp) apply(rule trans) apply(rule_tac p="(x ↔ y)" in supp_perm_eq[symmetric]) apply(rule fresh_star_supp_conv) apply(simp add: flip_def supp_swap fresh_star_def f1 f2) apply(simp add: flip_commute p) apply(simp add: Abs1_eq_iff) done lemma Abs_lst_fcb: fixes xs ys :: "'a :: fs" and S T :: "'b :: fs" assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)" and f1: "⋀x. x ∈ set (ba xs) ⟹ x ♯ f xs T" and f2: "⋀x. ⟦supp T - set (ba xs) = supp S - set (ba ys); x ∈ set (ba ys)⟧ ⟹ x ♯ f xs T" and eqv: "⋀p. ⟦p ∙ T = S; p ∙ ba xs = ba ys; supp p ⊆ set (ba xs) ∪ set (ba ys)⟧ ⟹ p ∙ (f xs T) = f ys S" shows "f xs T = f ys S" using e apply - apply(subst (asm) Abs_eq_iff2) apply(simp add: alphas) apply(elim exE conjE) apply(rule trans) apply(rule_tac p="p" in supp_perm_eq[symmetric]) apply(rule fresh_star_supp_conv) apply(drule fresh_star_perm_set_conv) apply(rule finite_Diff) apply(rule finite_supp) apply(subgoal_tac "(set (ba xs) ∪ set (ba ys)) ♯* f xs T") apply(metis Un_absorb2 fresh_star_Un) apply(subst fresh_star_Un) apply(rule conjI) apply(simp add: fresh_star_def f1) apply(simp add: fresh_star_def f2) apply(simp add: eqv) done lemma Abs_set_fcb: fixes xs ys :: "'a :: fs" and S T :: "'b :: fs" assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)" and f1: "⋀x. x ∈ ba xs ⟹ x ♯ f xs T" and f2: "⋀x. ⟦supp T - ba xs = supp S - ba ys; x ∈ ba ys⟧ ⟹ x ♯ f xs T" and eqv: "⋀p. ⟦p ∙ T = S; p ∙ ba xs = ba ys; supp p ⊆ ba xs ∪ ba ys⟧ ⟹ p ∙ (f xs T) = f ys S" shows "f xs T = f ys S" using e apply - apply(subst (asm) Abs_eq_iff2) apply(simp add: alphas) apply(elim exE conjE) apply(rule trans) apply(rule_tac p="p" in supp_perm_eq[symmetric]) apply(rule fresh_star_supp_conv) apply(drule fresh_star_perm_set_conv) apply(rule finite_Diff) apply(rule finite_supp) apply(subgoal_tac "(ba xs ∪ ba ys) ♯* f xs T") apply(metis Un_absorb2 fresh_star_Un) apply(subst fresh_star_Un) apply(rule conjI) apply(simp add: fresh_star_def f1) apply(simp add: fresh_star_def f2) apply(simp add: eqv) done lemma Abs_res_fcb: fixes xs ys :: "('a :: at_base) set" and S T :: "'b :: fs" assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)" and f1: "⋀x. x ∈ atom ` xs ⟹ x ∈ supp T ⟹ x ♯ f xs T" and f2: "⋀x. ⟦supp T - atom ` xs = supp S - atom ` ys; x ∈ atom ` ys; x ∈ supp S⟧ ⟹ x ♯ f xs T" and eqv: "⋀p. ⟦p ∙ T = S; supp p ⊆ atom ` xs ∩ supp T ∪ atom ` ys ∩ supp S; p ∙ (atom ` xs ∩ supp T) = atom ` ys ∩ supp S⟧ ⟹ p ∙ (f xs T) = f ys S" shows "f xs T = f ys S" using e apply - apply(subst (asm) Abs_eq_res_set) apply(subst (asm) Abs_eq_iff2) apply(simp add: alphas) apply(elim exE conjE) apply(rule trans) apply(rule_tac p="p" in supp_perm_eq[symmetric]) apply(rule fresh_star_supp_conv) apply(drule fresh_star_perm_set_conv) apply(rule finite_Diff) apply(rule finite_supp) apply(subgoal_tac "(atom ` xs ∩ supp T ∪ atom ` ys ∩ supp S) ♯* f xs T") apply(metis Un_absorb2 fresh_star_Un) apply(subst fresh_star_Un) apply(rule conjI) apply(simp add: fresh_star_def f1) apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys") apply(simp add: fresh_star_def f2) apply(blast) apply(simp add: eqv) done lemma Abs_set_fcb2: fixes as bs :: "atom set" and x y :: "'b :: fs" and c::"'c::fs" assumes eq: "[as]set. x = [bs]set. y" and fin: "finite as" "finite bs" and fcb1: "as ♯* f as x c" and fresh1: "as ♯* c" and fresh2: "bs ♯* c" and perm1: "⋀p. supp p ♯* c ⟹ p ∙ (f as x c) = f (p ∙ as) (p ∙ x) c" and perm2: "⋀p. supp p ♯* c ⟹ p ∙ (f bs y c) = f (p ∙ bs) (p ∙ y) c" shows "f as x c = f bs y c" proof - have "supp (as, x, c) supports (f as x c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) then have fin1: "finite (supp (f as x c))" using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) have "supp (bs, y, c) supports (f bs y c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) then have fin2: "finite (supp (f bs y c))" using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) obtain q::"perm" where fr1: "(q ∙ as) ♯* (x, c, f as x c, f bs y c)" and fr2: "supp q ♯* ([as]set. x)" and inc: "supp q ⊆ as ∪ (q ∙ as)" using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"] fin1 fin2 fin by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) have "[q ∙ as]set. (q ∙ x) = q ∙ ([as]set. x)" by simp also have "… = [as]set. x" by (simp only: fr2 perm_supp_eq) finally have "[q ∙ as]set. (q ∙ x) = [bs]set. y" using eq by simp then obtain r::perm where qq1: "q ∙ x = r ∙ y" and qq2: "q ∙ as = r ∙ bs" and qq3: "supp r ⊆ (q ∙ as) ∪ bs" apply(drule_tac sym) apply(simp only: Abs_eq_iff2 alphas) apply(erule exE) apply(erule conjE)+ apply(drule_tac x="p" in meta_spec) apply(simp add: set_eqvt) apply(blast) done have "as ♯* f as x c" by (rule fcb1) then have "q ∙ (as ♯* f as x c)" by (simp add: permute_bool_def) then have "(q ∙ as) ♯* f (q ∙ as) (q ∙ x) c" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm1) using inc fresh1 fr1 apply(auto simp add: fresh_star_def fresh_Pair) done then have "(r ∙ bs) ♯* f (r ∙ bs) (r ∙ y) c" using qq1 qq2 by simp then have "r ∙ (bs ♯* f bs y c)" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm2[symmetric]) using qq3 fresh2 fr1 apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) done then have fcb2: "bs ♯* f bs y c" by (simp add: permute_bool_def) have "f as x c = q ∙ (f as x c)" apply(rule perm_supp_eq[symmetric]) using inc fcb1 fr1 by (auto simp add: fresh_star_def) also have "… = f (q ∙ as) (q ∙ x) c" apply(rule perm1) using inc fresh1 fr1 by (auto simp add: fresh_star_def) also have "… = f (r ∙ bs) (r ∙ y) c" using qq1 qq2 by simp also have "… = r ∙ (f bs y c)" apply(rule perm2[symmetric]) using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) also have "... = f bs y c" apply(rule perm_supp_eq) using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) finally show ?thesis by simp qed lemma Abs_res_fcb2: fixes as bs :: "atom set" and x y :: "'b :: fs" and c::"'c::fs" assumes eq: "[as]res. x = [bs]res. y" and fin: "finite as" "finite bs" and fcb1: "(as ∩ supp x) ♯* f (as ∩ supp x) x c" and fresh1: "as ♯* c" and fresh2: "bs ♯* c" and perm1: "⋀p. supp p ♯* c ⟹ p ∙ (f (as ∩ supp x) x c) = f (p ∙ (as ∩ supp x)) (p ∙ x) c" and perm2: "⋀p. supp p ♯* c ⟹ p ∙ (f (bs ∩ supp y) y c) = f (p ∙ (bs ∩ supp y)) (p ∙ y) c" shows "f (as ∩ supp x) x c = f (bs ∩ supp y) y c" proof - have "supp (as, x, c) supports (f (as ∩ supp x) x c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh inter_eqvt supp_eqvt) then have fin1: "finite (supp (f (as ∩ supp x) x c))" using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) have "supp (bs, y, c) supports (f (bs ∩ supp y) y c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh inter_eqvt supp_eqvt) then have fin2: "finite (supp (f (bs ∩ supp y) y c))" using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) obtain q::"perm" where fr1: "(q ∙ (as ∩ supp x)) ♯* (x, c, f (as ∩ supp x) x c, f (bs ∩ supp y) y c)" and fr2: "supp q ♯* ([as ∩ supp x]set. x)" and inc: "supp q ⊆ (as ∩ supp x) ∪ (q ∙ (as ∩ supp x))" using at_set_avoiding3[where xs="as ∩ supp x" and c="(x, c, f (as ∩ supp x) x c, f (bs ∩ supp y) y c)" and x="[as ∩ supp x]set. x"] fin1 fin2 fin apply (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) done have "[q ∙ (as ∩ supp x)]set. (q ∙ x) = q ∙ ([as ∩ supp x]set. x)" by simp also have "… = [as ∩ supp x]set. x" by (simp only: fr2 perm_supp_eq) finally have "[q ∙ (as ∩ supp x)]set. (q ∙ x) = [bs ∩ supp y]set. y" using eq by(simp add: Abs_eq_res_set) then obtain r::perm where qq1: "q ∙ x = r ∙ y" and qq2: "(q ∙ as ∩ supp (q ∙ x)) = r ∙ (bs ∩ supp y)" and qq3: "supp r ⊆ (bs ∩ supp y) ∪ q ∙ (as ∩ supp x)" apply(drule_tac sym) apply(simp only: Abs_eq_iff2 alphas) apply(erule exE) apply(erule conjE)+ apply(drule_tac x="p" in meta_spec) apply(simp add: set_eqvt inter_eqvt supp_eqvt) done have "(as ∩ supp x) ♯* f (as ∩ supp x) x c" by (rule fcb1) then have "q ∙ ((as ∩ supp x) ♯* f (as ∩ supp x) x c)" by (simp add: permute_bool_def) then have "(q ∙ (as ∩ supp x)) ♯* f (q ∙ (as ∩ supp x)) (q ∙ x) c" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm1) using inc fresh1 fr1 apply(auto simp add: fresh_star_def fresh_Pair) done then have "(r ∙ (bs ∩ supp y)) ♯* f (r ∙ (bs ∩ supp y)) (r ∙ y) c" using qq1 qq2 apply(perm_simp) apply simp done then have "r ∙ ((bs ∩ supp y) ♯* f (bs ∩ supp y) y c)" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm2[symmetric]) using qq3 fresh2 fr1 apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) done then have fcb2: "(bs ∩ supp y) ♯* f (bs ∩ supp y) y c" by (simp add: permute_bool_def) have "f (as ∩ supp x) x c = q ∙ (f (as ∩ supp x) x c)" apply(rule perm_supp_eq[symmetric]) using inc fcb1 fr1 apply (auto simp add: fresh_star_def) done also have "… = f (q ∙ (as ∩ supp x)) (q ∙ x) c" apply(rule perm1) using inc fresh1 fr1 by (auto simp add: fresh_star_def) also have "… = f (r ∙ (bs ∩ supp y)) (r ∙ y) c" using qq1 qq2 apply(perm_simp) apply simp done also have "… = r ∙ (f (bs ∩ supp y) y c)" apply(rule perm2[symmetric]) using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) also have "... = f (bs ∩ supp y) y c" apply(rule perm_supp_eq) using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) finally show ?thesis by simp qed lemma Abs_lst_fcb2: fixes as bs :: "atom list" and x y :: "'b :: fs" and c::"'c::fs" assumes eq: "[as]lst. x = [bs]lst. y" and fcb1: "(set as) ♯* f as x c" and fresh1: "set as ♯* c" and fresh2: "set bs ♯* c" and perm1: "⋀p. supp p ♯* c ⟹ p ∙ (f as x c) = f (p ∙ as) (p ∙ x) c" and perm2: "⋀p. supp p ♯* c ⟹ p ∙ (f bs y c) = f (p ∙ bs) (p ∙ y) c" shows "f as x c = f bs y c" proof - have "supp (as, x, c) supports (f as x c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) then have fin1: "finite (supp (f as x c))" by (auto intro: supports_finite simp add: finite_supp) have "supp (bs, y, c) supports (f bs y c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) then have fin2: "finite (supp (f bs y c))" by (auto intro: supports_finite simp add: finite_supp) obtain q::"perm" where fr1: "(q ∙ (set as)) ♯* (x, c, f as x c, f bs y c)" and fr2: "supp q ♯* Abs_lst as x" and inc: "supp q ⊆ (set as) ∪ q ∙ (set as)" using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"] fin1 fin2 by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) have "Abs_lst (q ∙ as) (q ∙ x) = q ∙ Abs_lst as x" by simp also have "… = Abs_lst as x" by (simp only: fr2 perm_supp_eq) finally have "Abs_lst (q ∙ as) (q ∙ x) = Abs_lst bs y" using eq by simp then obtain r::perm where qq1: "q ∙ x = r ∙ y" and qq2: "q ∙ as = r ∙ bs" and qq3: "supp r ⊆ (q ∙ (set as)) ∪ set bs" apply(drule_tac sym) apply(simp only: Abs_eq_iff2 alphas) apply(erule exE) apply(erule conjE)+ apply(drule_tac x="p" in meta_spec) apply(simp add: set_eqvt) apply(blast) done have "(set as) ♯* f as x c" by (rule fcb1) then have "q ∙ ((set as) ♯* f as x c)" by (simp add: permute_bool_def) then have "set (q ∙ as) ♯* f (q ∙ as) (q ∙ x) c" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm1) using inc fresh1 fr1 apply(auto simp add: fresh_star_def fresh_Pair) done then have "set (r ∙ bs) ♯* f (r ∙ bs) (r ∙ y) c" using qq1 qq2 by simp then have "r ∙ ((set bs) ♯* f bs y c)" apply(simp only: fresh_star_eqvt set_eqvt) apply(subst (asm) perm2[symmetric]) using qq3 fresh2 fr1 apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) done then have fcb2: "(set bs) ♯* f bs y c" by (simp add: permute_bool_def) have "f as x c = q ∙ (f as x c)" apply(rule perm_supp_eq[symmetric]) using inc fcb1 fr1 by (auto simp add: fresh_star_def) also have "… = f (q ∙ as) (q ∙ x) c" apply(rule perm1) using inc fresh1 fr1 by (auto simp add: fresh_star_def) also have "… = f (r ∙ bs) (r ∙ y) c" using qq1 qq2 by simp also have "… = r ∙ (f bs y c)" apply(rule perm2[symmetric]) using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) also have "... = f bs y c" apply(rule perm_supp_eq) using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) finally show ?thesis by simp qed lemma Abs_lst1_fcb2: fixes a b :: "atom" and x y :: "'b :: fs" and c::"'c :: fs" assumes e: "[[a]]lst. x = [[b]]lst. y" and fcb1: "a ♯ f a x c" and fresh: "{a, b} ♯* c" and perm1: "⋀p. supp p ♯* c ⟹ p ∙ (f a x c) = f (p ∙ a) (p ∙ x) c" and perm2: "⋀p. supp p ♯* c ⟹ p ∙ (f b y c) = f (p ∙ b) (p ∙ y) c" shows "f a x c = f b y c" using e apply(drule_tac Abs_lst_fcb2[where c="c" and f="λ(as::atom list) . f (hd as)"]) apply(simp_all) using fcb1 fresh perm1 perm2 apply(simp_all add: fresh_star_def) done lemma Abs_lst1_fcb2': fixes a b :: "'a::at_base" and x y :: "'b :: fs" and c::"'c :: fs" assumes e: "[[atom a]]lst. x = [[atom b]]lst. y" and fcb1: "atom a ♯ f a x c" and fresh: "{atom a, atom b} ♯* c" and perm1: "⋀p. supp p ♯* c ⟹ p ∙ (f a x c) = f (p ∙ a) (p ∙ x) c" and perm2: "⋀p. supp p ♯* c ⟹ p ∙ (f b y c) = f (p ∙ b) (p ∙ y) c" shows "f a x c = f b y c" using e apply(drule_tac Abs_lst1_fcb2[where c="c" and f="λa . f ((inv atom) a)"]) using fcb1 fresh perm1 perm2 apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt) done end