theory CallArityEnd2End imports ArityTransform CoCallAnalysisImpl begin locale CallArityEnd2End begin sublocale CoCallAnalysisImpl. lemma fresh_var_eqE[elim_format]: "fresh_var e = x ⟹ x ∉ fv e" by (metis fresh_var_not_free) lemma example1: fixes e :: exp fixes f g x y z :: var assumes Aexp_e: "⋀a. Aexp e⋅a = esing x⋅(up⋅a) ⊔ esing y⋅(up⋅a)" assumes ccExp_e: "⋀a. CCexp e⋅a = ⊥" assumes [simp]: "transform 1 e = e" assumes "isVal e" assumes disj: "y ≠ f" "y ≠ g" "x ≠ y" "z ≠ f" "z ≠ g" "y ≠ x" assumes fresh: "atom z ♯ e" shows "transform 1 (let y be App (Var f) g in (let x be e in (Var x))) = let y be (Lam [z]. App (App (Var f) g) z) in (let x be (Lam [z]. App e z) in (Var x))" proof- from arg_cong[where f = edom, OF Aexp_e] have "x ∈ fv e" by simp (metis Aexp_edom' insert_subset) hence [simp]: "¬ nonrec [(x,e)]" by (simp add: nonrec_def) from `isVal e` have [simp]: "thunks [(x, e)] = {}" by (simp add: thunks_Cons) have [simp]: "CCfix [(x, e)]⋅(esing x⋅(up⋅1) ⊔ esing y⋅(up⋅1), ⊥) = ⊥" unfolding CCfix_def apply (simp add: fix_bottom_iff ccBindsExtra_simp) apply (simp add: ccBind_eq disj ccExp_e) done have [simp]: "Afix [(x, e)]⋅(esing x⋅(up⋅1)) = esing x⋅(up⋅1) ⊔ esing y⋅(up⋅1)" unfolding Afix_def apply simp apply (rule fix_eqI) apply (simp add: disj Aexp_e) apply (case_tac "z x") apply (auto simp add: disj Aexp_e) done have [simp]: "Aheap [(y, App (Var f) g)] (let x be e in Var x)⋅1 = esing y⋅((Aexp (let x be e in Var x )⋅1) y)" by (auto simp add: Aheap_nonrec_simp ABind_nonrec_eq pure_fresh fresh_at_base disj) have [simp]: "(Aexp (let x be e in Var x)⋅1) = esing y⋅(up⋅1)" by (simp add: env_restr_join disj) have [simp]: "Aheap [(x, e)] (Var x)⋅1 = esing x⋅(up⋅1)" by (simp add: env_restr_join disj) have 1: "1 = inc⋅0" apply (simp add: inc_def) apply transfer apply simp done have [simp]: "Aeta_expand 1 (App (Var f) g) = (Lam [z]. App (App (Var f) g) z)" apply (simp add: 1 del: exp_assn.eq_iff) apply (subst change_Lam_Variable[of z "fresh_var (App (Var f) g)"]) apply (auto simp add: fresh_Pair fresh_at_base pure_fresh disj intro!: flip_fresh_fresh elim!: fresh_var_eqE) done have [simp]: "Aeta_expand 1 e = (Lam [z]. App e z)" apply (simp add: 1 del: exp_assn.eq_iff) apply (subst change_Lam_Variable[of z "fresh_var e"]) apply (auto simp add: fresh_Pair fresh_at_base pure_fresh disj fresh intro!: flip_fresh_fresh elim!: fresh_var_eqE) done show ?thesis by (simp del: Let_eq_iff add: map_transform_Cons map_transform_Nil disj[symmetric]) qed end end