theory LetRecB+ −
imports "../Nominal2"+ −
begin+ −
+ −
atom_decl name+ −
+ −
nominal_datatype let_rec:+ −
trm =+ −
Var "name"+ −
| App "trm" "trm"+ −
| Lam x::"name" t::"trm" bind x in t+ −
| Let_Rec bp::"bp" t::"trm" bind "bn bp" in bp t+ −
and bp =+ −
Bp "name" "trm"+ −
binder+ −
bn::"bp \<Rightarrow> atom list"+ −
where+ −
"bn (Bp x t) = [atom x]"+ −
+ −
thm let_rec.distinct+ −
thm let_rec.induct+ −
thm let_rec.exhaust+ −
thm let_rec.fv_defs+ −
thm let_rec.bn_defs+ −
thm let_rec.perm_simps+ −
thm let_rec.eq_iff+ −
thm let_rec.fv_bn_eqvt+ −
thm let_rec.size_eqvt+ −
+ −
+ −
lemma Abs_lst_fcb2:+ −
fixes as bs :: "'a :: fs"+ −
and x y :: "'b :: fs"+ −
and c::"'c::fs"+ −
assumes eq: "[ba as]lst. x = [ba bs]lst. y"+ −
and fcb1: "(set (ba as)) \<sharp>* f as x c"+ −
and fresh1: "set (ba as) \<sharp>* c"+ −
and fresh2: "set (ba bs) \<sharp>* c"+ −
and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"+ −
and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"+ −
and props: "eqvt ba" "inj ba"+ −
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 \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and + −
fr2: "supp q \<sharp>* ([ba as]lst. x)" and + −
inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"+ −
using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" and x="[ba as]lst. x"] + −
fin1 fin2+ −
by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+ −
have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp+ −
also have "\<dots> = [ba as]lst. x"+ −
by (simp only: fr2 perm_supp_eq)+ −
finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp+ −
then obtain r::perm where + −
qq1: "q \<bullet> x = r \<bullet> y" and + −
qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and + −
qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 qq4: "q \<bullet> as = r \<bullet> bs" using qq2 props unfolding eqvt_def inj_on_def+ −
apply(perm_simp)+ −
apply(simp)+ −
done+ −
have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)+ −
then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"+ −
by (simp add: permute_bool_def)+ −
then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"+ −
apply(simp add: 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 \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 qq4+ −
by simp+ −
then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"+ −
apply(simp add: 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 (ba bs)) \<sharp>* f bs y c" by (simp add: permute_bool_def)+ −
have "f as x c = q \<bullet> (f as x c)"+ −
apply(rule perm_supp_eq[symmetric])+ −
using inc fcb1 fr1 by (auto simp add: fresh_star_def)+ −
also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" + −
apply(rule perm1)+ −
using inc fresh1 fr1 by (auto simp add: fresh_star_def)+ −
also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq4 by simp+ −
also have "\<dots> = r \<bullet> (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 max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"+ −
by (simp add: permute_pure)+ −
+ −
nominal_primrec+ −
height_trm :: "trm \<Rightarrow> nat"+ −
and height_bp :: "bp \<Rightarrow> nat"+ −
where+ −
"height_trm (Var x) = 1"+ −
| "height_trm (App l r) = max (height_trm l) (height_trm r)"+ −
| "height_trm (Lam v b) = 1 + (height_trm b)"+ −
| "height_trm (Let_Rec bp b) = max (height_bp bp) (height_trm b)"+ −
| "height_bp (Bp v t) = height_trm t"+ −
--"eqvt"+ −
apply (simp only: eqvt_def height_trm_height_bp_graph_def)+ −
apply (rule, perm_simp, rule, rule TrueI)+ −
--"completeness"+ −
apply (case_tac x)+ −
apply (case_tac a rule: let_rec.exhaust(1))+ −
apply (auto)[4]+ −
apply (case_tac b rule: let_rec.exhaust(2))+ −
apply blast+ −
apply(simp_all)+ −
apply (erule_tac c="()" in Abs_lst_fcb2)+ −
apply (simp_all add: fresh_star_def pure_fresh)[3]+ −
apply (simp add: eqvt_at_def)+ −
apply (simp add: eqvt_at_def)+ −
apply(simp add: eqvt_def)+ −
apply(perm_simp)+ −
apply(simp)+ −
apply(simp add: inj_on_def)+ −
--"The following could be done by nominal"+ −
apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])+ −
apply (simp add: meta_eq_to_obj_eq[OF height_bp_def, symmetric, unfolded fun_eq_iff])+ −
apply (subgoal_tac "eqvt_at height_bp bp")+ −
apply (subgoal_tac "eqvt_at height_bp bpa")+ −
apply (subgoal_tac "eqvt_at height_trm b")+ −
apply (subgoal_tac "eqvt_at height_trm ba")+ −
apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bp)")+ −
apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bpa)")+ −
apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl b)")+ −
apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl ba)")+ −
defer+ −
apply (simp add: eqvt_at_def height_trm_def)+ −
apply (simp add: eqvt_at_def height_trm_def)+ −
apply (simp add: eqvt_at_def height_bp_def)+ −
apply (simp add: eqvt_at_def height_bp_def)+ −
apply (subgoal_tac "height_bp bp = height_bp bpa")+ −
apply (subgoal_tac "height_trm b = height_trm ba")+ −
apply simp+ −
apply (subgoal_tac "(\<lambda>as x c. height_trm (snd (bp, b))) as x c = (\<lambda>as x c. height_trm (snd (bpa, ba))) as x c")+ −
apply simp+ −
apply (erule_tac c="()" in Abs_lst_fcb2)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: eqvt_at_def)+ −
apply (simp add: eqvt_at_def)+ −
defer defer+ −
apply (subgoal_tac "(\<lambda>as x c. height_bp (fst (bp, b))) as x c = (\<lambda>as x c. height_bp (fst (bpa, ba))) as x c")+ −
apply simp+ −
apply (erule_tac c="()" in Abs_lst_fcb2)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: fresh_star_def pure_fresh)+ −
apply (simp add: eqvt_at_def)+ −
apply (simp add: eqvt_at_def)+ −
--""+ −
done+ −
+ −
termination by lexicographic_order+ −
+ −
end+ −
+ −
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+ −