Theory Nominal2.Nominal2_FCB

```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(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)
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(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)
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(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)
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(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(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")
apply(blast)
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(blast)
done
have "as ♯* f as x c" by (rule fcb1)
then have "q ∙ (as ♯* f as x c)"
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
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
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)
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)"
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
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
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(blast)
done
have "(set as) ♯* f as x c" by (rule fcb1)
then have "q ∙ ((set as) ♯* f as x c)"
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
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
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
```