Quelle  IVSubst.thy

(*<*)[]\^><>v]\^supv) = (x1 ↔ c1" using assms flip_subst_vby auto
theoryIVSubst
  imports Syntax
begin
  (*>*)

chapter \<>Variable

text ‹[[atom z1]]lst. c1 = [ato z2]]lst. c2" 
java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

  ‹ (c1[z1::=[x]^v) = c1'[z1'::=[x']^v"

  has_subst_v = fs +
 fixes subst_v :: "'a::fs ==> x ==> v ==> 'a::fs" (‹ c1[z1::=[x]^\^sub>v" using assms fresh by simp
 assumes fresh_subst_v_if: "y ♯ a ∧ y ♯sharp> v ∧ (y ♯ a ∨ y = atom x))"
 and forget_subst_v[simp]: "atom x ♯ a ==> subst_v a x v = a"
 and subst_v_id[simp]: "subst_v a x (V_var x) = a"
 and eqvt[simp,eqvt]: "(p::perm) ∙ (subst_v a x v) = (subst_v (p ∙ a) (p ∙x) (p ∙
 and flip_subst_v[simp]: "atom x ♯ c ==> ((x ↔ z) ∙ ".. ==c1[z1^v" ussubst_v_si fresh_prod4 ass by auto
java.lang.NullPointerException
 

  subst_v_flfinally show ?thes by auto
 fixes z1::x z2::x and x1::x and x2:::x
 assumes "[[atom z1]]lst. c1 = [[atom z2]]lst. c2"
 and<>Values
java.lang.NullPointerException
  -
 v] z1) ∙
 moreover have "(c2[z2::=[x1] x ==> v" where
 ultimately show ?thesis using Abs1_eq_iff_all(3)[of z1 c1 z"subst_vv (V_lit l) x v = V_lit l"
  (ei bseq_iff_freeh(3) flip_commute)
 

  subst_v_flip_eq_two:
 fixes z1::x and z2::x and x1::x and x2::x
 assumes "[[atom z1]]lst. c1 = [[atom z2]]lst. c2"
 shows "(c1[z1::=b]_{V_cons}tyid c v') x v = tyid c (su v' x v)"
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 obtain x::x where *:"atom x ♯ (auto ip: ev_e ubt_v_rp_ef,metis v.stn_xas)
 hence "(c1[z1::=[x] x ==> v" (‹ [1000,50,50] 1000)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 thus ?thesis using subst_v_simple_commute * fresh_prod4 by metis
 

  subst_v_flip_eq_three:
 assumes "[[atom z1]]lst. c1 = [[atom z1']]lst. c1'" and "atom x ♯ c1" and "atom x' ♯: v.induct) (auto simp: fresh_Pair)
java.lang.NullPointerException
  -
 have "atom x' ♯ c1[z1::=[x]^v]_v" using assms fresh_subst_v_if by simp
 hence "(x ↔ x') ∙ (c1[z1::=[x]^v]_v) = c1[z1::=[x]^v]_v[x::=[x'] su[simp]:
 also have "... = c1[z1::=[x'] <> v"
 also have "... = c1'[z1'::=[x']^v" uigsubst_v_lip_eqne[oz1 1 z'c'x'] g assms byaut
 finally show ?thesis by auto
  asssapyinutvrl:.nduc

 

  ‹ = subst_vv"

 
 subst_vv :: "v \<show  
 "subst_vv (V_lit l) x v = V_lit l"
  "subst_vv (V_var y) x v = (if x = y then v else V_var y)"
  "subst_vv (V_cons tyid c v') x v = V_cons tyid c (subst_vv v' x v)"
  "subst_vv (V_consp tyid c b v') x v = V_consp tyid c b (subst_vv v' x v)"
  "subst_vv (V_pair v1 v2) x v = V_pair (subst_vv v1 x v ) (subst_vv v2 x v )"
 by(auto simp: eqvt_def subst_vv_graph_aux_def, metis v.strong_exhaust)
  (eqvt) by lexicographic_order

 
 subst_vv_abbrev :: "v ==> x ==> v ==> v" (‹
 where
 "v[x::=']\<^>v≡

 ubst_vv_if [simps_vv_if [simp]:
 "j ♯
 using supp_l_empty apply (induct t rule: v.induct,auto simp add: subst_vv.simps fresh_def, auto)
 by (simp add: supp_at_base |met

  forget_sushow "p∙ t1) (p ∙ v)"
 by (induct tm rule: v.induct) (simp_all add: fresh_at_base)

java.lang.NullPointerException
 by (induct tm rule: v.induct) simp_all

  [simp]:
 "atom j ♯ c ==> (↔ c) = c[z::=[x]^v"
 y(induct t levidc) (at smp:feh_Pair

  suf xxan: :x
 atom j <>< j ==>v_vv = tm[j::=u]_vv[i::=t]_vv"
 by (induct tm rule: v.induct) auto

  subst_vv_var_flip[simp]:
 fixes v::v
 assumes "atom y ♯ v"
 shows "(y ↔ x) ∙
 using assms apply(induct v rule:v.induct)
 apply auto
 using l.fresh l.perm_simps l.strong_exhaust supp_l_empty permute_pure permute_list.simps fresh_def flip_fresh_fresh apply fastforce
 using permute_pure apply blast+
 done

  v :: has_sui subst_e: "e \<> 
 

 
 "subst_v = subst_vv"

  proof
 fix j::atom and i::x and x::v and t::v
 show "(j ♯ l v') ) x v = ( (A ( v' x v) )"
 using fresh_subst_vv_if[of j t i x] subst_v_v_def by metis

 fix a::x and tm::v and x::v
 showshow a a \sharp tm \<> 
 using forget_subst_vv subst_v_v_def by simp

 fix a::x and tm::v
 show "subst_v tm a (V_var a) = tm" using subst_vv_id subst_v_v_def by simp

 fix p::perm and x1::x and v::v and t1::v
 show "p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)"
 using subst_v_v_def by simp

 fix x::x and c::v and z::x
 show "atom x ♯ ( (AE_appP fb v' ) xv = ( (AE_a f b (sbst_ v' x )) )"
 using subst_v_v_def by simp

 fix x::x and c::v and z::x
 show "atom x ♯ c ==> c[z::=[x]^v] v2 x v)) )")"
 using subst_v_v_def by simp
 

 

  ‹Expressions›

  subst_ev :: "e ==> x ==> v ==> e" where
 "subst_ev ( (AE_val v') ) x v = ( (AE_val (subst_vv v' x v)) )"
  "subst_ev ( (AE_app f v') ) x v = ( (AE_app f (subst_vv v' x v )) )"
  "subst_ev ( (AE_appP f b v') ) x v = ( (AE_appP f b (subst_vv v' x v )) )"
  "subst_ev ( (AE_op opp v1 v2) ) x v = ( (AE_op opp (subst_vv v1 x v ) (subst_vv v2 x v )) )"
  "subst_ev [#1 v']^e x v = [#1 (subst_vv v' x v )]^e"
  "subst_ev [#2 v']^e x v = [#2 (subst_vv v' x v )]^e"
  "subst_ev ( (AE_mvar u)) x v = AE_mvar u"
  "subst_ev [| v' |]^e xv = [1 (substvv ' x v )]\^sup>e""
  "subst_ev ( AE_concat v1 v2) x v = AE_concat (subst_vv v1 x v ) (subst_vv v2 x v )"
  "subst_ev ( AE_split v1 v2) x v = AE_split (subst_vv v1 x v ) (subst_vv v2 x v )"
 by(simp add: eqvt_def subst_ev_graph_aux_def,auto)(meson e.strong_exhaust)

  (eqvt) by lexicographic_order

 
java.lang.NullPointerException
 where
 "e[x::=v']_ev ≡su ( (AEu))x v A u"

  size_subst_ev [simp]: "size ( subst_ev A i x) = size A"
 apply (nominal_induct A avoiding: i x rule: e.strong_induct)
 by auto

  forget_subst_ev [simp]: "atom a ♯ A ==> subst_ev A a x = A"
 apply (nominal_induct A avoiding: a x rule: e.strong_induct)
 by (auto simp: fresh_at_base)

  subst_ev_id [simp]: "subst_ev A a (V_var a) = A"
 by (nominal_induct A avoiding: a rule: e.strong_induct) (auto simp: fresh_at_base)

  fresh_subst_ev_if [simp]:
 "j ♯ (subst_ev A i x ) = ((atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i)))"
 apply (induct A rule: e.induct)
 unfolding subst_ev.simps fresh_subst_vv_if apply auto+
 using pure_fresh fresh_opp_all apply metis+
 done

  subst_ev_commute [simp]:
 "atom j ♯ A ==> (A[i::=t]_ev)[j::=u]_ev = A[i::=t[j::=u]_vv]_ev"
 by (nominal_induct A avoiding: i j t u rule: e.strong_induct) (auto simp: fresh_at_base)

  subst_ev_var_flip[simp]:
 fixes e::e and y::x and x::x
 assumes "atom y ♯ e"
 shows "(y ↔ x) ∙ e = e [x::=V_var y]_ev"
 using assms apply(nominal_induct e rule:e.strong_induct)
 apply (simp add: subst_v_v_def)
 apply (metis (mono_tags, lifting)
 apply (metis (mono_tags, lifting) b.eq_iff b.perm_simps e.fresh e.perm_simps flip_b_id subst_ev.simps subst_vv_var_flip)
 
 apply (rule_tac y=x in opp.strong_exhaust)
 using subst_vv_var_flip flip_def by (simp add: flip_def permute_pure)+
 using subst_vv_var_flip flip_def by (simp add: flip_def permute_pure)+

  subst_ev_flip:
 fixes e::e and ea::e and c::x
 assumes "atom c ♯
java.lang.NullPointerException
  -
 have "e[x::=v'] ==>Rightarrow> e" (‹
 also have "... = ((c ↔ x) ∙ e)[c::=v']_ev" using subst_ev_var_flip assms by simp
 also have "... = ((c ↔ xa) ∙ ea)[c::=v']_ev" using assms flip_commute by metis
 also have "... = ea[xa::=v']_ev" using subst_ev_var_flip assms by simp
 finally show ?thesis by auto
 

  subst_ev_var[simp]:
 "(AE_val (V_var x))[x::=[z]^v]_ev = AE_val (V_var z)"
 by auto

  e :: has_subst_v
 

 
 "subst_v = subst_ev"

  proof
 fix j::atom and i::x and x::v and t::e
 show "(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯
 using fresh_subst_ev_if[of j t i x] subst_v_e_def by metis

 fix a::x and tm::e and x::v
 show "atom a ♯ tm ==> subst_v tm a x = tm"
 using forget_subst_ev subst_v_e_def by simp

 fix a::x and tm::e
 show "subst_v tm a (V_var a) = tm" using subst_ev_id subst_v_e_def by simp

 fix p::perm and x1::x and v::v and t1::e
 show "p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙
 using subst_ev_commute subst_v_e_def by simp

 fix x::x and c::e and z::x
java.lang.NullPointerException
 using subst_v_e_def by simp

 fix x::x and c::e and z::x
java.lang.NullPointerException
 using subst_v_e_def by simp
 
 

  subst_ev_commute_full:
 fixes e::e and w::v and v::v
 assumes "atom z ♯ v" and "atom x ♯ w" and "x ≠ z"
java.lang.NullPointerException
 using assms by(nominal_induct e rule: e.strong_induct,simp+)

  subst_ev_v_flip1[simp]:
 fixes e::e
 assumes "atom z1 ♯ (z,e)" and "atom z1' ♯ (z,e)"
 shows"(z1 ↔ z1') ∙ e[z::=v]_e nominal_in A avoi: a a :strong_induct) (auto : f)
 using assms proof(nominal_induct e rule:e.strong_induct)
  (simp add: flip_def fresh_Pair swap_fresh_fresh)+

  ‹

  subst_cev :: "ce ==> x ==>A \andj\<> 
 "subst_cev ( (CE_val v') ) x v = ( (CE_val (subst_vv v' x v )) )"
  "subst_cev ( (CE_op opp v1 v2) ) x v = ( (CE_op opp (subst_cev v1 x v ) (subst_cev v2 x v )) )"
  "subst_cev ( (CE_fst v')) x v = CE_fst (subst_cev v' x v )"
  "subst_cev ( (CE_snd v')) x v = CE_snd (subst_cev v' x v )"
  "subst_cev ( (CE_len v')) x v = CE_len (subst_cev v' x v )"
  "subst_cev ( CE_concat v1 v2) x v = CE_concat (subst_cev v1 x v ) (subst_cev v2 x v )"
 apply (simp add: eqvt_def subst_cev_graph_aux_def,auto)
 by (meson ce.strong_exhaust)

  (eqvt) by lexicographic_order

 
 subst_cev_abbrev :: "ce ==> x ==> v ==> ce" (‹_[_::=_]_cev› [1000,50,50] 500)
 where
 "e[x::=v']_cev ≡ subst_cev e x v'"

  size_subst_cev [simp]: "size ( subst_cev A i x ) = size A"
 by (nominal_induct A avoiding: i x rule: ce.strong_induct,auto)

  forget_subst_cev [simp]: "atom a ♯ A ==> subst_cev A a x = A"
 by (nominal_induct A avoiding: a x rule: ce.strong_induct, auto simp: fresh_at_base)

  subst_cev_id [simp]: "subst_cev A a (V_var a) = A"
 by (nominal_induct A avoiding: a rule: ce.strong_induct) (auto simp: fresh_at_base)

  fresh_subst_cev_if [simp]:
 "j ♯ (subst_cev A i x ) = ((atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i)))"
 (nominal_induct A avoiding: i x rule: ce.strong_induct)
 case (CE_op opp v1 v2)
 then show ?case using fresh_subst_vv_if subst_ev.simps e.supp pure_fresh opp.fresh
 fresh_e_opp
 using fresh_opp_all by auto
 (auto)+

  subst_cev_commute [simp]:
 "atom j ♯ A ==> (subst_cev (subst_cev A i t ) j u) = subst_cev A i (subst_vv t j u )"
 by (nominal_induct A avoiding: i j t u rule: ce.strong_induct) (auto simp: fresh_at_base)

  subst_cev_var_flip[simp]:
 fixes e::ce and y::x and x::x
 assumes "atom y ♯ e"
 shows "(y ↔ x) ∙ e = e [x::=V_var y]_cev"
 using assms proof(nominal_induct e rule:ce.strong_induct)
 case (CEnduct A rule: e.nduct)
 then show ?case using subst_vv_var_flip by auto
 
 case (CE_op opp v1 v2)
 hence yf: "aty ♯ blast
 have " (y ↔ x) ∙ (CE_op opp v1 v2 ) = CE_op ((y ↔ x) ∙ opp) ( (y ↔ x) ∙ v1 ) ( (y ↔ x) ∙ v2)"
  opp.perm_simps ce.p.perm_simps permute_pure ce.fres opp.st by pr
java.lang.NullPointerException
 by (simp add: CE_op.hyps(1) CE_op.hyps(2))
 finally show ?case using subst_cev.simps opp.perm_simps opp.strong_exhaust
  (me (full_types))
 ( (auto simp add: permute_pure subst_vv_var_flip)+)

  subst_cev_flip:
 fixes e::ce and ea::ce and c::x
 assumes "atom c ♯ (e, ea)" and "atom c ♯ (x, xa, e, ea)" and "(x ↔ c) ∙nom A rule: e.stro) (a sim fresh_atbas
java.lang.NullPointerException
  -
 have "e[x::=v']_:andy:: x::x
 also have "... = ((c ↔ x) ∙
java.lang.NullPointerException
 also have "... = ea[xa::=v']_cev" using subst_ev_var_flip assms by simp
 finally show ?thesis by auto
 

  subst_cev_var[simp]:
 fixes z::x and x::x
 shows "[[x]^v]
 by auto

  ce :: has_subst_v
 

 
 "subst_v = subst_cev"

  proof
 fix j::atom and i::x and x::v anapply (metis mono_t, lifting) b.eq bperm_simps e.fresh e. flip_b_id substev. substvv_var_)
 show "(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯s)
 using fresh_subst_cev_if[of j t i x] subst_v_ce_def by metis

 fix a::x and tm::ce and x::v
 show "atom a ♯ tm ==> subst_v tm a x = tm"
 using forget_subst_cev subst_v_ce_def by simp

 fix a::x and tm::ce
 show "subst_v tm a (V_var a) = tm" using subst_cev_id subst_v_ce_def by simp

 fix p::perm and x1::x and v::v and t1::ce
 show "p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)"
 using subst_cev_commute subst_v_ce_def by simp

 fix x::x and c::ce and z::x
 show "atom x ♯ c ==> ((x ↔ for x
 using subst_v_ce_def by simp

 fix x::x and c::ce and z::x
 show "atom x ♯ c ==> c [z::=V_var x]\<^    using
 using subst_v_ce_def by simp
 

 

  subst_cev_commute_full:
 fixes e::ce and w::v and v::v
 assumes "atom z ♯ v" and "atom x ♯ w" and "x ≠ z"
java.lang.NullPointerException
 using assms by(nominal_induct e rule: ce.strong_induct,simp+)


  subst_cev_v_flip1[simp]:
 fixes e::ce::ce
 assumes "atom z1 ♯ (z,e)" and "atom z1' ♯ (z,e)"
java.lang.NullPointerException
 using assms apply(nominal_induct e rule:ce.strong_induct)
 by (simp add: flip_def fresh_Pair swap_fresh_fresh)+

  ‹Constraints›

  subst_cv :: "c ==> x ==> v ==>
 "subst_cv (C_true) x v = C_true"
  "subst_cv (C_false) x v = C_false"
  "subst_cv (C_conj c1 c2) x v = C_conj (subst_cv c1 x v ) (subst_cv c2 x v )"
  "subst_cv (C_disj c1 c2) x v = C_disj (subst_cv c1 x v ) (subst_cv c2 x v )"
  "subst_cv (C_imp c1 c2) x v = C_imp (subst_cv c1 x v ) (subst_cv c2 x v )"
  "subst_cv (e1 == e2) x v = ((subst_cev e1 x v ) == (subst_cev e2 x v ))"
  "subst_cv (C_not c) x v = C_not (subst_cv c x v )"
 apply (simp add: eqvt_def subst_cv_graph_aux_def,auto)
 using c.strong_exhaust by metis
  (eqvt) by lexicographic_order

 
 subst_cv_abbrev :: "c ==> x ==> v ==> c" (‹
 where
 "c[x::=v']_cv ≡ subst_cv c x v'"

  size_subst_cv [simp]: "size ( subst_cv A i x ) = size A"
 by (nominal_induct A avoiding: i x rule: c.strong_induct,auto)

  forget_subst_cv [simp]: "atom a ♯ A ==> subst_cv A a x = A"
 by (nominal_induct A avoiding: a x rule: c.strong_induct, auto simp: fresh_at_base)

  subst_cv_id [simp]: "subst_cv A a (V_var a) = A"
 by (nominal_induct A avoiding: a rule: c.strong_induct) (auto simp: fresh_at_base)

  fresh_subst_cv_if [simp]:
 "j ♯ (subst_cv A i x ) ⟷ (atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i))"
 by (nominal_induct A avoiding: i x rule: c.strong_induct, (auto simp add: pure_fresh)+)

 subst_cv_commute [simp]:
java.lang.NullPointerException
 by (nominal_induct A avoiding: i j t u rule: c.strong_induct) (auto simp: fresh_at_base)

  let_s_size [simp]: "size s ≤ size (AS_let x e s)"
 apply (nominal_induct s avoiding: e x rule: s_branch_s_branch_list.strong_induct(1))
 apply auto
 done

  subst_cv_var_flip[simp]:
 fixes c::c
 assumes "atom y ♯ c"
 shows "(y ↔ x) ∙ c = c[x::=V_var y]_cv"
  by(nominal_induct c rule:c.,(simp add flipsubst_v_ce_def)+)

  c :: has_subst_v
 

 
 "subst_v = subst_cv"

  proof
 fix j::atom and i::x and x::v and t::c
 show "(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))"
 using fresh_subst_cv_if[of j t i x] subst_v_c_def by metis

 fix a::x and tm::c and x::v
 showsubst_ev_var[imp:
 using forget_subst_cv subst_v_c_def by simp

 fix a::x and tm::c
 show "subst_v tm a (V_var a) = tm" using subst_cv_id subst_v_c_def by simp

 p:perand x1::x a v::v and t1::::c
 show "p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙
 using subst_cv_commute subst_v_c_def by simp

 fix x::x and c::c and z::x
 show "atom x ♯
 using subst_cv_var_flip subst_v_c_def by simp

 and c::c and ::
java.lang.NullPointerException
 using subst_cv_var_flip subst_v_c_def by simp
 

 

  subst_cv_var_flip1[simp]:
 fixes c::c
 assumes "atom y ♯ c"
 shows "(x ↔ y) ∙ c = c[x::=V_var y]_c<sharp> t \< j
 using subst_cv_var_flip flip_commute
 by (metis assms)

  subst_cv_v_flip3[simp]:
 fixes c::c
 assumes "atom z1 ♯ c" and "atom z1' ♯ c"
 shows"(z1 ↔ z1') ∙ c[z::=[z1]^v]_cv = c[z::=[z1']^v]_cv"
  -
 consider "z1' = z" | "z1 = z" | "atom z1 ♯ z ∧ atom z1' ♯ z" by force
 then show ?thesis proof(cases)
 case 1
 then show ?thesis using 1 assms by auto
 next
 case 2
 then show ?thesis using 2 assms by auto
 next
 case 3
 then show ?thesis using assms by auto
 qed
 

  subst_cv_v_flip[simp]:
 fixes c::c
 assumes "atom x ♯ c"
 shows "((x ↔ z) ∙ c)[x::=v]_cv = c [z::=v]_cv"
 using assms subst_v_c_def by auto

  subst_cv_commute_full:
 fixes c::c
 assumes "atom z ♯ v" and "atom x ♯ w" and "x≠z"
 shows "(c[z::=w]_cv)[x::=v]_cv = (c[x::=v]_c[of j t i x sub bym
 using assms proof(nominal_induct c rule: c.strong_induct)
 ase ( (C_eq e1e1 e2
 then show ?case using subst_cev_commute_full by simp
 (force+)

  subst_cv_eq[simp]:
 assumes "atom z1 ♯ e1"
 shows "(CE_val (V_var z1) == e1 )[z1::=[x] = tm"
  -
 have "?A = (((CE_val (V_var z1))[z1::=[x]^v]_cev) == e1)" using subst_cv.simps assms by simp
 thus ?thesis by simp
 

  ‹

  ‹
  x is fresh in v but this causes problems with proofs.›

  subst_gv :: "Γ \  "su tm a (Vvar a) tm" us s by s
 "subst_gv GNil x v = GNil"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 goal_cases)
 case 1
 then show ?case by(simp add: eqvt_def subst_gv_graph_aux_def )
 
 case (3 P x)
 then show ?case by (metis neq_GNil_conv prod_cases3)
 (fast+)
  (eqvt) by lexicographic_order

 
 subst_gv_abbrev :: "Γ ==> x ==>
 where
java.lang.NullPointerException

 size_subst_gv [simp]: "size ( G i x ) \<> 
 by (induct G,auto)

  forget_subst_gv [simp]: "atom a ♯ G ==> subst_gv G a x = G"
 apply (induct G ,auto)
 using fresh_GCons fresh_PairD(1) not_self_fresh apply blast
 apply (simp add: fresh_GCons)+
 done

  fresh_subst_gv: "atom a ♯ G ==>
 (induct G)
 case GNil
 then show ?case by auto
 
 case (GCons xbc G)
java.lang.NullPointerException
 show ?case proof(cases "x=x'")
 case True
 have "atom a ♯ G" using GCons fresh_GCons by bla simp
 thus ?thesis using subst_gv.simps(2)[of x' b' c' G] GCons xbc True
 next
 case False
 then show ?thesis using subst_gv.simps(2)[
 qed
 

  subst_gv_flip:
 fixes x::x anx and xxa::x and z::x and c::c and b::b and Γ
 assumes "atom xa ♯ ((x, b, c[z::=[x]^"atom z \sharp" and "atomx <>w
java.lang.NullPointerException
  -
 have "(x ↔ xa) ∙ ((x, b, c[z::=[x]^v]_cv) #_\Γ Γ) = (( (x ↔ xa) ∙ x, b, (x ↔ xa) ∙ c[z::=[x]^v]_cv) #_\Γ ((x ↔e r: e.,si+)
 using subst Cons_eqvt flip_fresh using G_cons_flip by si
java.lang.NullPointerException
 also have "... = ((xa, b, c[z::=V_var xa]_ze)"
 also have "... = ((xa, b, c[z::=V_var xa]_cv) #_\Γ Γ)" using assms flip_fresh_fresh by blast
 finally show ?thesis by simp
 

  ‹Types›

  subst_tv :: "τ ==> x ==> v ==> τ rule:estrong_induct)
 "atom z ♯ (x,v) ==> subst_tv { z : b | c }
 apply (simp add: eqvt_def subst_tv_graph_aux_def )
 apply auto
 subgoal for P a aa b
 apply(rule_tac y=a and c="(aa,b)" in τ.strong_exhaust)
 by (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)
 apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)
  -
 fix z :: x and c :: c and za :: x and xa :: x and va :: v and ca :: c and cb :: x
 assume a1: "atom za ♯ va" and a2: "atom
 assume a4: "atom cb ♯
 assume a10:"cb ≠ xa"
 note assms = a10 a9 a8 a7 a6 a5 a4 a3 a2 a1

 have "c[z::=V_var cb]_cv = ca[za::=V_var cb]_cv" using assms by auto
 hence "c[z::=V_var cb] v' x' x v )) )) )
 moreover have "c[z::=V_var cb]_cv[xa::=va]_cv = c[xa::=va]_cv[z::=V_var cb]_cv" using subst_cv_commute_full[of z va xa "V_var cb" ] assms fresh_def v.supp by fastforce
 moreover have "ca[za::=V_var cb] "subst_cev ( (CE_op opp v1 v2) ) x v = ( (CE_op opp (sub v1 x v ) subst_ v2x v )) )"
 using subst_cv_commute_full[of za va xa "V_var cb" ] assms fresh_def v.supp by fastforce

 ultimately show "c[xa::=va]_cv[z::=V_var cb]_cv = ca[xa::=va]₎ x =CE_ (su v x v )"
 

  (eqvt) by lexicographic_order

 
java.lang.NullPointerException
 where
 "t[x::=v]_\τ_v ≡ subst_tv t x v"

  size_subst_tv [simp]: "size ( subst_tv A i x ) = size A"
  ((nominal_in A avoi: i x r: \<tau.
 case (T_refined_type x' b' c')
 then show ?case by auto
 

  forget_subst_tv [simp]: "atom a ♯ A ==> subst_tv A a x = A"
 apply (nominal_induct A avoiding: a x rule: τ.strong_induct)
 apply(auto simp: fresh_at_base)
 one

  subst_tv_id [simp]: "subst_tv A a (V_var a) = A"
 by (nominal_induct A avoiding: a rule: τasimp:fresh_at_base)

  fresh_subst_tv_if [simp]:
 "j ♯ (subst_tv A i x ) ⟷ (atom i ♯st)
 apply (nominal_induct A avoiding: i x rule: τ.strong_induct)
 using fresh_def supp_b_empty x_fresh_b by auto

  subst_tv_commute [simp]:
 atom y \sharp τt]_<au>_sub🚫
 by (nominal_induct τ avoiding: x y t v rule: τ.strong_induct) (auto simp: fresh_at_base)

  subst_tv_var_flip [simp]:
 fixes x::x and xa::x and τ::τ
 assumes "atom xa ♯ τ"
 shows "(x ↔ xa) ∙ τ = τ[x::=V_var xa]_\τ_v"
  -
 obtain z::x and b and c where zbc: "atom z ♯ (x,xa, V_var xa) ∧ τ = { z : b | c }"
 using obtain_fresh_z by (metis prod.inject subst_tv.cases)
 hence "atom xa ∉ supp c - { atom z }" using τ.supp[of z b c] fresh_def supp_b_empty assms
 by auto
 moreover have "xa ≠ z" using zbc fresh_prod3 by force
 ultimately have xaf: "atom xa ♯ c" using fresh_def by auto
 have "(x ↔ xa) ∙ τ = { z : b | (x ↔ xa) ∙ c }"
 by (metis τ.perm_simps empty_iff flip_at_base_simps(3) flip_fresh_fresh fresh_PairD(1) fresh_PairD(2) fresh_def not_self_fresh supp_b_empty v.fresh(2) zbc)
 also have "... = { z : b | c[x::=V_var xa]_cv }" using subst_cv_v_flip xaf
 by (metis permute_flip_cancel permute_flip_cancel2 subst_cv_var_flip)
 finally show ?thesis using subst_tv.simps zbc
 using fresh_PairD(1) not_self_fresh by force
 

  τ :: has_subst_v
 

 
 "subst_v = subst_tv"

  proof
 fix j::atom and i::x and x::v and t::τ
 show "(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))"

 proof(nominal_induct t avoiding: i x rule:τ.strong_induct)
 case (T_refinedabbreviation
 hence " j ♯ {x\Rightarrow v <Rightarrow ]_,] 500)
 also have "... = (atom i ♯ { z : b | c } ∧ j ♯ { z : b | c } ∨ j ♯ x ∧ (j ♯ { z : b | c } ∨ j = atom i))"
 unfolding τ.fresh using subst_v_c_def fresh_subst_v_if
 using T_refined_type.hyps(1) T_refined_type.hyps(2) x_fresh_b by auto
 finally s
 qed

 fix a::x and tm::τ and x::v
 show "atom a ♯ tm ==> subst_v tm a x = tm"
 apply(nominal_induct tm avoiding: a x rule:τ.strong_induct)
  subst forget_subst_v sub.simps subsτ

 fix a::x and tm::τ
 show "subst_v tm a (V_var a) = tm"
 apply(nominal_induct tm avoiding: a rule:τ.strong_induct)
 using subst_v_c_def forget_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by simp

  and v::v and t1::τ
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 apply(nominal_induct tm avoiding: a x rule:τ)(aut simp:
 using subst_v_c_def forget_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by simp

 fix x::x and c::τ and z::x
java.lang.NullPointerException
 apply(nominal_induct c avoiding: z x rule:τ.strong_induct)
 using subst_v_c_def flip_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by auto

 fix x::x and c::τ and z::x
 show "atom x ♯ c ==> c[z::=[x]^v]_v[x::=v]_v = c[z::=v]_v"
 apply(nominal_induct c avoiding: x v z rule:τ.strong_induct)
 using subst_v_c_def subst_tv.simps subst_v_τ_def fresh_Pair   j ♯
 by (metis flip_commute subst_tv_commute subst_tv_var_flip subst_v_τ_def subst_vv.simps(2))
 

 

  subst_tv_commute_full:
 fixes c::τ
 assumes "atom z ♯ v" and "atom x ♯ w" and "x≠z"
 shows "(c[z::=w]_\τ_v)[x::=v]_\τ_v = (c[x::=v]_\τ_v)[z::=w]_\τ_v"
 using assms proof(nominal_induct c avoiding: x v z w rule: τ.strong_induct)
 case (T_refined_type x1a x2a x3a)
 then show ?case using subst_cv_commute_full by simp
 

  type_eq_subst_eq:
 fixes v::v and c1::c
 assumes "{ z1 : b1 | c1 } = { z2 : b2 | c2 }"
 shows "c1[z1::=v]_cv = c2[z2::=v]_cv"
 using subst_v_flip_eq_two[of z1 c1 z2 c2 v] τ.eq_iff assms subst_v_c_def by simp

  ‹Extract constraint from a type. e.strong_
  alpha-equivalent types give different answers ›

  c_of :: "τ ==> x ==>show ?case usingsub.simps e.supp pure_fresh opp.fresh
java.lang.NullPointerException
 (goal_cases)
 case 1
 then show ?case using eqvt_def c_of_graph_aux_def by force
 
 case (2 x y)
 then show ?case using eqvt_def c_of_graph_aux_def by force
 
 case (3 P x)
 then obtain x1::τ
 obtain z' and b' and c' where "x1 = { subst_cev_commute [simp]:
 then show ?case using 3 * by auto
 
 case (4 z1 x1 b1 c1 z2 x2 b2 c2)
 then show ?case using subst_v_flip_eq_two τ.eq_iff by (metis prod.inject type_eq_subst_eq)
 

  (eqvt) by lexicographic_order

  c_of_eq:
  "c_of {> x = c"
 (nominal_induct "{ x : b | c }" avoiding: x rule: τ.strong_induct)
 case (T_refined_type x' c')
 moreover hence "c_of { x' : b | c' } x = c'[x'::=V_var x]_cv" using c_of.simps by auto
 moreover have "{ x' : b | c' } = {e::ce and y::x and x::x
 moreover have "c'[x'::=V_var x]_cassumes "atom y ♯
 by (metis subst_cv_id)
 ultimately show ?case by auto
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 3

  obtain_fresh_z_c_ofe
 t:::"'b:fs"
 obtains z where "atom z ♯ t ∧ τ = { z : b_of τ | c_of τ z }"
  -
 obtain z and c where "atom z ♯ yf: "atom y \sharpv ∧ by bl
 moreover hence "c = c_of τ z" using c_of.simps using c_of_eq by metis
 ultimately show ?thesis
 using that by auto
 

  c_of_fresh:
 fixes x::x
 assumes "atom x ♯ (t,z)"
 shows "atom x ♯ c_of t z"
  -
  z' and c' where z"t = {| c' \<> 
 hence *:"c_of t z = c'[z'::=V_var z]_cv" using c_of.simps fresh_Pair by metis
 have "(atom x ♯ c' ∨ atom x ∈ set [atom z']) ∧ atom x ♯ b_of t" using τ.fresh assms z fresh_Pair by metis
 hence "atom x ♯ c'" using fresh_Pair z fresh_at_base(2) by fastforce
 moreover have "atom x ♯ V_var z" using assms fresh_Pair v.fresh by metis  opp.perm_simps ce.permpermute_pure ce.fre op.strong_exhaust by presburger
 ultimately show ?thesis using assms fresh_subst_v_if[of "atom x" c' z' "V_var z"] subst_v_c_def * by metis
 

  c_of_switch:
  z::x
 assumes "atom z ♯ t"
 shows "(c_of t z)[z::=V_var x]_simpsopp.perm_simps opp.strong_exhaust
  -
 obtain z' and c' where z:"t = { z' : b_of t | c' } ∧ atom z' ♯ (x,z)"
 hence "(atom z ♯( (auto simp add: permu subst_vv_var_fl)+)
 moreover have " atom z ∉ set [atom z']" using z fresh_Pair by force
 ultimately have **:"atom z ♯

java.lang.NullPointerException
 also have "... = c'[z'::=V_var x] and c::x
 finally show ?thesis using c_of.simps[of z' x "b_of t" c'] fresh_Pair z by metis
 

  type_eq_subst_eq1:  "atom c ♯> (x, , e, ea)" and " "(x \leftrightarrow ) <> 
 fixes v::v and c1::c
 assumes "{ z1 : b1 | c1 } = ({ z2 : b2 | c2 })" and "atom z1 ♯ c2"
 shows "c1[z1::=v]_cv = c2[z2::=v]_cv" and "b1=b2" and " c1 = (z1 ↔ z2) ∙ c2"
  -
 show "c1[z1::=v]_cv = c2[z2::=v]_']_\^sub>🚫
 show "b1=b2" using τ.eq_iff assms by blast
 have "z1 = z2 ∧ c1 = c2 ∨ z1
 using τ.eq_iff Abs1_eq_iff[of z1 c1 z2 c2] assms by blast
 thus "c1 = (z1 \ ↔<bullet 
 

  type_eq_subst_eq2:
 fixes v::v and c1::c
 assumes "{also hav"... = (c 🚫
 shows "c1[z1::=v]_cv = c2[z2::=v]_cv" and "b1=b2" and "[[atom z1]]lst. c1 = [[atom z2]]lst. c2"
  -
 show "c1[z1::=v]_cv = c2[z2::=v]_cv" using type_eq_subst_eq assms by blast
 show "b1=b2" using τ.eq_iff assms by blast
 show "[[atom z1]]lst. c1 = [[atom z2]]lst. c2"
 using τ.eq_iff assms by auto
 

 type_eq_subst_eq3:
 fixes v::v and c1::c
 assumes "{ z1 : b1 | c1 } = ({ z2 : b2 | c2 })" and "atom z1 ♯ c2"
java.lang.NullPointerException
 using type_eq_subst_eq1 assms subst_v_c_def
 by (metis subst_cv_var_flip)+

  type_eq_flip:
 assumes "atom x ♯ c"
 shows "{ z : b | c } = { x : b | (x ↔ z ) ∙qe
 using τ.
 by (metis (no_types, lifting) flip_fresh_fresh)

 c_of_true:
 "c_of { z' : B_bool | TRUE } x = C_true"
 (nominal_induct "{ z' : B_bool | TRUE }" avoiding: x rule:τ.strong_induct)
 case (T_refined_type x1a x3a)
 hence "{ z' : B_bool | TRUE } fix z::x and x::x
 then show ?case using subst_cv.simps c_of.simps T_refined_type
 type_eq_subst_eq3
 by (metis type_e)
 

 type_eq_subst:
 assumes "atom x ♯ c"
 shows "{ z : b | c } = { x : b | c[z::=[x]^{instantiation} ce :: has_subst_v
 using τ.eq_iff Abs1_eq_iff assms
 using subst_cv_var_flip type_eq_flip by auto

  type_e_subst_fresh:
 fixes x::x and z::x
 assumes "atom z ♯ (x,v)" and "atom x ♯ e"
 shows "{
 using assms subst_tv.simps subst_cv.simps forget_subst_cev by simp

 
 fixes x::x and z::x
 assumes "atom z ♯ (x,v)" and "atom x ♯s> subst_v t i x) = ((atom i \sharp t <and ♯or j = atom i)))"
 shows "{ z : b | CE_val (V_var z) == CE_val v' }[x::=v]_\τ_v = { z : b | CE_val (V_var z) == CE_val v' }"
 using assms subst_tv.simps subst_cv.simps by simp

  subst_tbase_eq:
 "b_of τ = b_of τ[x::=v]_\τ_v"
  -
 obtain z and b and c where zbc: "τ = { z:b|c} ∧ atom z ♯ (x,v)" using τ.exhaust
 by (metis prod.inje subst_tv.c
 hence "b_of { z:b|c} = b_of { z:b|c}[x::=v]_\τ_v" using subst_tv.simps by simp
 thus ?thesis using zbc by blast
 

  subst_tv_if:
 assumes "atom z1 ♯subst_v tm a = tm"
 shows "{ z1 : b | CE_val (v'[x::=v]_vv) == CE_val (V_lit l) IMP (c'[x::=v]_cv)[z'::=[z1]^v]_simp
 { z1 : b | CE_val v' == CE_val (V_lit l) IMP
 using subst_cv_commute_full[of z' v x
 by simp

  substubst_tv_tid:
 assumes "atom za ♯ (x,v)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using assms subst_tv.simps subst_cv.simps by presburger


  b_of_subst:
 "b_of (τ[x::=v]_show "p ∙subst_v (p \<> 
  -
 obtain z b c where *:"τ = { z : b | c } ∧ atom z ♯ (x,v)" using obtain_fresh_z by metis
 thus ?thesis using subst_tv.simps * by auto
 

  subst_tv_flip:
 assumes "τ]_v τ ""atom x ♯"atom x' ♯
 shows "((x' ↔ x) ∙ τ')[x'::=v]_\τ_v = τ"
  -
 have "(x' ↔ x) ∙ v = v ∧ (x' ↔ x) ∙ τ = τ" using assms flip_fresh_fresh by auto
 thus ?thesis using subst_tv.eqvt[of "(x' ↔
 

  subst_cv_true:
 "{ z : B_id tid | TRUE } = { z : B_id tid | TRUE }[x::=v]_sharp>c ==> ∙v"
  -
 obtain za::x where "atom za ♯ (x,v)" using obtain_fresh by auto
 hence "{ z : B_id tid | TRUE } = { za: B_id tid | TRUE }" using τ.eq_iff Abs1_eq_iff by fastforce
 moreover have "{ za : B_id tid | TRUE } = { za : B_id tid | TRUE }[x::=v]_\τ_v"
 using subst_cv.simps subst_tv.simps by (simp add: ‹
 show ?thesis by ar
 

  t_eq_supp:
 assumes "({"atatom \sharp c ==>:=V_var x]\<^<sub
 shows "supp c - { atom z } = supp c1 - { atom z1 }"
  -
 have "supp c - { atom z } ∪ supp b = supp c1 - { atom z1 } ∪ supp b1" using τ.supp assms
 by (metis list.set(1) list.simps(15) sup_bot.right_neutral supp_b_empty)
 moreover have "supp b = supp b1" using assms τ.eq_iff by simp
 moreover have "atom z1 ∉ supp b1 ∧ atom z ∉ supp b" using supp_b_empty by simp
 using subst_v_ce_def by simp
 by (metis τ.eq_iff τ.supp
 

  fresh_t_eq:
 fixes x::x
 assumes "({ z : b | c }
 s "atom x ♯
  -
 have "supp c - { atom z } ∪ supp b = supp cc - { atom zz } ∪fixes e::ce and w::v and v::v
 by (metis list.set(1) list.simps(15) sup_bot.right_neutral supp_b_empty)
 moreover have "atom x ∉ "atom z ♯> w" and "x ≠
 ultimately have "atom x ∉ supp cc - { atom zz } ∪ supp b" by force
 hence "atom x ∉ supp cc" using assms by simp
java.lang.NullPointerException
 

  ‹Mutable Variable Context›

  subst_dv :: "Δ ==> x ==> v ==> Δ" where
 "subst_dv DNil x v = DNil"
  "subst_dv ((u,t) #_\Δ Δ) x v = ((u,t[x::=v] using assms by(nominal_induct e rule: cc.strong_induc,simp+)
 apply (simp add: eqvt_def subst_dv_graph_aux_def,auto )
 using delete_aux.elims by (metis Δ
  (eqvt) by lexicographic_order

 
 subst_dv_abbrev :: "Δ subst_cev_v_f[simp]:
 where
 "Δ:ce

  dmap :: "(u*τ ==> u*τ) ==> Δ ==> Δ" where
 "dsumes "atom z1 <sharp 
  "dmap f ((u,t)#_\ΔΔ) = (f (u,t) #_<Delta> (dmap f Δ ))"
 apply (simp add: eqvt_def dmap_graph_aux_def,auto )
 shows"(z1"(z1 ↔bullet> e[z::=v_z:: ((z1 \<leftrightarrow > v)]_"
  (eqvt) by lexicographic_order

  subst_dv_iff:
 "Δ[x::=v]_Delta\<lambda(tx::=v]\^su>τv)) Δ"
 by(induct Δ, auto)

  size_subst_dv [simp]: "size ( subst_dv G i x) ≤ size G"
 by (induct G,auto)

  forget_subst_dv [simp]: duct e rule:ce.strong_induct)
 duct Gato)
 using fresh_DCons fresh_PairD(1) not_self_fresh apply fastforce
  (simp add: fresh_DCons)+ )+
 done

  subst_dvget_s[simp: aom x♯ simp]] "ubst_v (V_a ) = a"
  "(u,\>) ∈
java.lang.NullPointerException
java.lang.NullPointerException

  fresh_subst_dv:
 fixes x::x
 assumes "atom xa ♯"
 shows "atom xa ♯Δ[x::=v]_\<|  
 using assms proof(induct Δ rule:Δ_induct)
  DNil
 bst_v_flip_eq_two:
  "(cz1::=b]\^sv) = (c2[z2::=b]_{subst_cv}bst_cv (C_d 1 c2c2) xx v _ (subs_vc x v ) (subst_cv c2 x v )"
 
 then show ?case usingsubx v = C_imp (subst_ c1 v c1x v (ubstc c2v )"java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 thus ?esi sn suubtv_simple_commte fehod4 metis

  fr "( ↔(subsb c x )"
 fixes j::atom and i::x and x::v and t::Δ
 "
 shows "(j ♯slso ". = c1[1::=['\^ssup>>v]\<^                     applyValues›
  assms proo(induct t rule: Δ
 Niljava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 then show ?case using subst_gv.si by(auto s:evt_def sb_vv_g
 
 case (DCons u' t' DD')
 then show ?case unfolding subst_dv.simps using fresh_DCons fresh_su
 

java.lang.NullPointerException

  ‹
java.lang.NullPointerException

  "case_sum (λ(ase_su<.Inl
 bst_svv : "s \\ x ==> v ==>
 and subst_branchv 🚫v"
 ubst_branchlv :"nch \Rightarrow> x ==> br
  (A_val v') ) x v = (A
  "atom y ♯ proof
  "atom y ♯ subst_v by ( (nominal_induct A avooiding: a x rrule: ng_indut, aut imp: f)
java.lang.NullPointerException
  "subst_sv (AS_assign y v') x v = AS_assign y (subst_vv v' x v )"
  (AS_if v' s1 s2)) ) x v = (AS_if_if (subbstvv v' x v ) (subst_ss xx v ) (subst_svs2 x v ) ))"  
  "atom u ♯ subst_sv (AS_var u τ (subst_tv τx v ) (subst_vv v' x v ) (ust_sv s x v ) "
 )x = A_hle (s(us_v s1 x ) (sub)"
  "subst_sv (AS_seq s1 s2) x v = AS_seq(us_vs x v) (subt_sv s2 x v "
  (AS_assert c s) x v AS_(sust_cv c x v) (sst_sv sx v)"
 ection \openExpressions›

 t_branchlv (AS_final c) v=A_finanal (subst_branhv c "
 bst_branchlv (AS_c
 apply (auto,sx v = ( (Eo op s_vv1 x v ) (sutvv xv ) )"
java.lang.NullPointerException

 have eqvt"< st,ong_indc sp add:pure_fres)+)
 eqvt_at (λa. projl (subst_sv_subst_branchv_subst_branchlv_sumC (Inl a))) (s, xa, va)"
 apply(simp add: eqvt_at_def)
 apply(rule)
 apply(subst Projl_permute)
 apply(thiin_ta+
 apply (simp add: subst_sv_subst_branchv
 
 apply (case_tac "Ex1Inl (s,,xava)))")
 lemma sizsimp]: ize ( subst_v A ix = size AA"
 
  simp: feha_base)
 apply simp
 apply(cases rule: subst_sv_subst_branchv_subst_branchlv_graph.cases)
 apply(assumption)
 apply(rule_tac x="Sum_Type.projl x" in exI,clarify,rule the1_equality,blast,simp (no_asm) only: sum.sel)+
 lemma ubst_e [simp]: "subst_ev A a (V_var a) a) = A"

 apply(simp)+
 done

 { 
  java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 then show ?case proof(cases x')
 case (Inl a) thus P
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 20
 e(e aa bbcc
 trong_exhaust ng_ehau fesh_sta_net by ts
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 next
 case (Inr b) thus P
 proof(cases b)
 case (Inl a) thus P proof(cases a)
 case (fields aa bb cc)
 en show ?thesis u usi Inr InInl l 1 s_brancbanch_lis.stongg_ehas fresh_s_ins byy metis
 qed
 

 case (fields aa bb cc)
  ?hsis usin n Inr2 s "((y \\leftrightarrow> x) ∙v"
 qed
 qed
 qed
 next
 case (2 y s ya xa va sa c)
 thus ?case using eqvt_triple eqvt_at_proj by blast
 next
 case 3 y 2 yax asa s2 )
 thus ?case u
 next
 =ub_v
  j::atm and i::x and x x::v and
 using fresh_Pair u_fresh_xvby auto
 ultimatelyfix :x and t:ead x:v
 next
 x1 1 x1a xa va s1a c)
 thus sho subst_ m a (a (V_vara) =tsing sub_v_id us___def b
 }how p ∙ t1) (p ∙ "ubst_v = ubs
 
 qvt by eioric_oder

 
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 where
 ^ub>v \equiv stsv s xs x v"

 
 subst_branchv_abbrev :: "> z1) ∙
 where
 >s subst_branchv s x v"

  \<>Expressions
  B and and C avoig ulesac__branch_list.stron_indu f a::xx adt::c and ::

  "sub ( (CE_len v' v')) v = CE_len (subst_cev v' x v )"
 by (nominal_induct A and B and C avoiding: a x rule: s_branch_sux_

  subst_sv_id [simp]: "subst_sv A a (V_var a) = A" and "subst_branchv B a (V_var a
  C aviding: rule: s_branchs_branch_list.strog_iut)
 case (A
 then shohow ?casse
 by (metis (no_types, "[:='\^sub>_ev ≡

 case (AS_match v branch_s)
 t_None_eq sbt_ev_id subts.simps sbst_sv.ps subst_ttv_id v.fresh ust_vv_id sh "subst_v t a(V_V_ar a) = sing subst_cvv_id subst_v_c_def by simp 
 by metis
 (auto)q

  fresh_subst_sv_if_rl:
 shows
 "( freshre
 
 "(atom x ♯<>subst_v
 apply(nominal_induct s and cs and css avoiding: v x rule: s_branch_s_branch_list.strong_induct)
 using pu

java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 shows "j ♯
 "j ♯
  (subst_branchlv css x v ) ==>
  " (y \<leftrightarrow :x an c::c and z::x
 branch list x s )
java.lang.NullPointerException
 
 s tosimpdd: pere_psub)+
 es atom ♯c_v =
  -
 y e s) x v = (AS_let y e s')" forget_subst_by simp
 hence "j ♯ (AS_let y e s')" using AS_let by argo
 show ?thesis Tru by blast
 next
 case False
 
java.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then show ?thesis using AS_let fresh_None fresh_subst_ev_if list.discI list.set_cases srnhsbrn
  fix a:x and tm::ce

 
 case (AS_let2 y τ s1 s2)
 thus ?case proof(1::x and v::v and :c
 case True
 using subst_cev_commute subst_v_ce_ef by s bysimp 
 show atom x ♯c ==> z) ∙
 then show ?thesis using True by blast

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 have "subst_> y) ∙
 then show ?thesis using AS_let2
 sh_subst_tv_ifsub st_tvtv_iif listst.discI s.set_cases s_branchbranchit.freresh(etiassms)
  <>  z1') \ e[z::=v]_esub> f c::
 uto)+

  fresh_subst_sv_if[simp]:
  x::xnd v::v
 shows "j \ (C_false) x v= C_false"
 " "ubst e = 2 x = ((subst_ e1 x v )= (substcev e2x
 ng rssubstsv_if_lr freh_sub by metis+

 
  A::sadt::v an x and i
 atom j\<harp thenr [simp]: "atom a ♯
 "atom j ♯
 atom j \>Lo> (subst_branchlv (subst_branchlv C i t) j u ) = subst_branchlv C i (subst_vv t ju )
 apply(nominal_induct A and B and C avoiding: i j t u rule: s_branch_s_branch_list.strong_induct)
 (auto simp: fresh_at_base)

 
 assumes " case 3
 shows "{size [simp]: "size e ?th?thesis using assms auto
 using τ
 by (metis Nominal2_Base.swa

  subst_sv_flip:
 fixes s::s and sa::s and v'::v
 assumes "atom c \<sharp( (v',x, xa, s, sa)" "atom x ♯ v'" and "(x \leftrightarrow c) ∙ c) ∙
java.lang.NullPointerException

 have "atom x ♯
  "a "atmc sharp> (s[x::=v'_<^sub>v, sa[xa::=v']_v)" using assms using fresh_subst_sv_if assms by( blast+ ,force)

 hence "s[x::=v'] subst_v t i x) = ((atom i ♯ j ♯ x ∧ ( \\sh> t ∨ j = atom i)))"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ve "... = ((xa \< c∙a by p
 also have "... = ((xa ↔ c) ∙ sa)[ ((xa ↔ xa) ::= ((xa ↔ v') ]_v" using assms
 by (mets flii_tsimps(1 flip_feh_freshfresh_PairD()
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 also ave "... = sax:=']\^>s_ssmsms breshfresh resh_Pair)
 finally show ?thesis by simp
 

  if_type_eq:
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 assumesshows "((x \<leftrightarrow =v]\^sub>c_>v"
java.lang.NullPointerException
 shows "({
  -
 have "atom z1' ♯v]_c\<>v
 moreover hence "(CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1']\<  showsv = c[z::=[z1']^using assms subst_v_c_def by by auto
 proof -
 have " consid "z1' = z" | "z1 = z "tom z1 ♯ subst_cv_commute:
 by auto
 also have "... = ((CE_val v == CE_val (V_lit ll)) IMP ((z1' ↔
 using ‹uto
 by (metis (mono_tags) ‹subt_cflip[sp]:
 also have "... = ((CE_val v == CE_val (V_lit ll)) IMP (ca[z\ s "c[z::=w]\^sub>c🚫
 using assms by fastforce
 finally show ?thesis by auto
 qed
 ultimately show ?thesis
 using \   "atom z ♯ w" and "x≠z"
java.lang.NullPointerException
 

  subst_sv_var_flip:
 fixes x::x and s::s and z::x
java.lang.NullPointerException
java.lang.NullPointerException
 "atom x ♯
 apply(nominal_induct s and cs and css avoiding: z x rule: s_branch_s_branch_list.strong_induct)
 using ext ‹
              apply (auto  ) (* This unpacks subst, perm *)
  ubst_tv_var_flip
    subst_v___java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
     defer
  gsub<Gamma> subst_gv g x v"
    defer 1
  using x_fresh_u apply blast
   defer 1
  using x_fresh_u Ab
   apply (simp add: subst_v_c_def)
  using x_fresh_u Abs1_eq_iff'(3) flip_fresh_fresh
  by (simp add: flip_fresh_fresh)

instantiationapply (inuct Gz1 \<> 
begin

definition
  "subst_v

instance
  fix j: vnd
  show  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    using 

  fix a::x and tm
    <> =tm
    usingst_sv

  fix a::x and tm:s
howt_v V_var bst_sv_idyp

   p::perm and x1::x and v::v and t1::s
  showbullet subst_v t1 x1 v  = _p\>
       subst_v_s_def

  fix x:x:sdx
  show "atom ed
    using subst_sv_var_flip subst_v_s_def by simp


  show "atom>c ==>v]java.lang.NullPointerException
      subst_v_s_def simp
qedbyo:t_def)
end

sectionz  andand  aandvca b:

nominal_function subst_ft_v :: "fun_typ ==> v ==>
  atom z <sharp> (x,v) \ <Longrightarrow> subst_ft_v ( AF_fun_typ z b c t (s::s)) x v = AF_fun_typ z b c[x::=v]_>\<>\
     apply(simp add: eqvt_def subst_ft_v_graph_aux_def ) )
    apply(simp add:fun_typ.strong_exhaust )
   apply(auto)
    apply(rule_tac y=a and c="(moreoverc_cccjava.lang.NullPointerException
    apply (auto

proof(goal_casesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  case  xaca
  hence  "c[
  (etis lp_comue sbt__vr_flip)
  hence "lemma imp"subst_tv bt_v A i x ) = siz A
  showcase)
    using _fip fip_mut by me
xt
  
  hence "t[byominal_induct:e<>trong_inductosimpbase
  hence  java.lang.NullPointerException
  then show ?case using subst_tv_commute_full 2
    by (metis atom_eq_iff fresh_atom fresh_atom_at_base v.fresh()
ed  then s ?case by byby(simpadd:evt_t_def subsst_gvv_gragrapph_aux_def )

nominal_termination (eqvt) by lexicographic_order

nominal_function subst_ftq_v :: "fun_typ_q x ==> fun_typ_q
  "atom bv ♯_idc <tau> avoiding: y t
| "   andnd::τshowaseisNil_convprod_cases3ases3
       apply(simpdeqvt_defubst_ftq_v_graph_aux_def
      apply(simp add:fun_typ_q.strong_exhaust )
     apply( -
   apply_ac=aa fun_typ_qust
    apply    usingzyetisprod.ses
proofbreviation
  case (1 bva<> z" using zbc fresh_prod3 by force
subst_ft_vsimps ms by (simp add: flip_fresh__fresh)
qed
nominal_termination (eqvt) by lexicograpr

lemma size_subst_ft[simp]: "size (subst_ft_v A x v) = size A"
  by(nominal_induct A avoiding: x v ru[x::=V_var xa]\<^bcv }"  using subst_cv_v_flip  where 

lemmausing_rDself_freshshorce
  by

lemma subst_ft_id
 inal_inducting typrong_induct

instantiation fun_typ :as_subst_vv
begin

definition 
  "subst_v = subst_ft_v"

instance proof

  fix j:lemma size_subst_gv] "size ( subst_g G i x ≤
  show "(j ♯
    apply(inal_inducttding xuleduct
    apply(simp only: subst_v_fun_typ_def subst_ft_v.simps )
usingtypesh_subst_v_if
    by auto

  fix a::andjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
grightarrowrrowsubst_v  
  (nominal_induct:rulerong_induct
    se_x1a java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
      applynductuto
  qed

  fix a: ndmn_typ
  showbst_vt_v )"
  proof(n fix x::x and c::::\tau>and z::x
    case (AF_fun_typ x1a x2a x3a x4a x5a)
java.lang.NullPointerException
  

  fix p::perm and x1::x and v::v an fix::x ad c:\tau and z::x
   " <ubst_v =t_v >)bullet(p\bullet"
  proof .strong_induct)
     2a45
    then show ?foldingst_v_fun_typ_defhusingsubst_v bst_ft_vimpsget_subst_svv_>def by fastforce
  qedubst_tv_commute_full

   :  _d:x
  show java.lang.NullPointerException
    case (T_efined_type x1x2a x3aa))
    by (a

java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  show "atom   x'andc c,singses3
    apply(
    apply auto
    bysimp _st_v_s_defst_v__tau subst_v_fun_typ_def
qed
end

instantiation: as_subst_v
begin

definition 
  "subst_v = subst_ftq_v"

instanceef
  fix j::atomcaseatom<  GConsbyblast
  show  "(j ♯
    plyninal_induct t at avoi: i x rule:fun_yp_.rong_induct,auto)
                   apply(auto simp add: subst_vfu_t_de us_vsubs_vv\tau>_def subst_v_fun_typ_q_dif )
bymetis (notpes) fesh_ubstv_ibstvfun_typ_def)+

i ad t::fun_typ__q and x::v
  show "atom \    ase
    apply(nominal_induct t avoiding: i x rule:fun_typ_q.strong_induct    then  ? using.()[of' Gnsxbcbymp
t_v_s_def>_def ubst_v_fun_typ_q_deffsubst_v_if

  fix:q
  showi
    moreoverflbrace x' : b | c' }
    simpddn_typ_defv_s_defst_v_τfixesx:: and:xandz:  c:: andd><>

  fixanddndp_q
  show "p ∙ subst_v t1 x1 v = subst_v (p ∙tom x \<sharpsharpsup>v]\^>c_ma)"nd < Gammand \Gamma  "atom x \<sharp > (z, c)"
    applyngyp_qinduct
    by(auto simp addproof

  fix moreoverc_of z" usiei
  show "
    apply(nominal_inductlemmaf_fresh
    sumes sharp (t,z)"

  fix x::x and c::fun_typ_q an ::x
  show "atom x ♯ z' : b_of  'rbrace∧ atom z' ♯
    (al_inductdingfun_typ_q
    (todbst_v_fun_typ_defdefv_s_def<tau_def subst_v_fun_typ_q_def fresh_subst_v_if
    by (metist_v_fun_typ_def___celbst_ft_vubst_v_simple_commutevrm_simpsmps
qed

endshowusingmsst_v_if''"V_var z" "ubst_v_c_def bby metis

java.lang.NullPointerException

lemmaa subst_v_fst_eq:
  " ` setD (Δ[x::=v]<sub>\</sub>Δv) = fst ` setD Δ"
 by(induct Δn subst C ons_eqvt flip_freshres sing G_con_lip by smp

  subst_gv_member_iff:
 fixes x'::x
 assumes "(x',b',c') <in >"
 shows "(x',b',c'[x::=v] x]_v" using subst_v_simple_commute subst_v_c_def assms c_oimps z *by mets
  -
 have "avassu\>z1 : b : b1 | c1 }b | c2 } zz1 ♯
 then show ?thesis using assms proof(induct Γshows "c1[z1[z1:=v]_\c=2" and "c1 = (z ↔ c2"
  GNil
 then show ?case by auto
 next
 case (GCons x1 b1 c1 Γ')
 eroof(cases "(x'b,' x,1,c1)")

java.lang.NullPointerException
 then show ?thesis using True by auto
 next Types\close
 case False
 have "x1≠.eq_iff Abs1_eq_iff[of z1 c1 z2 2 asss by bla> v \Rightarrowτ
 hence "(x', b', c') ∈ cc🚫 z1 : b1 | c1 } z2 : 2}
 moreover have "atom x ∉mdm Γ feshGCns
 ultimately have "(x', b', c'[x::=v]_Pa aa show "c1[z1:v]_<sb>c\subv" using type_eqsuseq assmss bllast
 ' c[:=]_\>toSet ((x1, b1, c1[x::=v]_v) #_Gamma> (Γ'[x::=v]_v))" by auto
 then show ?thesis using subst_gv.simps ‹fresh_at_basees_at_bas)
 qed
 qed
 

 resh_subst_gv_if:
 fixes j::atom and i::x and x::v and
 assumes "j ♯ t ∧ j ♯ x"
 shows "(j ♯ subst_gv t i x)"
 using assms proof(induct t rule: Γ_induct)
 case GNil
 then show ?case using subst_gv.simps fresh_GNil by auto
 
 case (GCons x' b' c' Γ')
 then show ?case unfolding subst_gv.simps using fresh_GCons fresh_subst_cv_if by auto
 

  ‹a6: "cb ≠" a "atom cb ♯"z≠

  set_GConsD: "y ∈c(Tefined_tytyexa 3
 o

 
 assumes c"
java.lang.NullPointerException
 using subst_gv.simps append_g.simps assms by auto