More on Function-defined subst.
theory Lambda
imports "../NewParser" Quotient_Option
begin
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam" bind_set x in l
thm lam.fv
thm lam.supp
lemmas supp_fn' = lam.fv[simplified lam.supp]
declare lam.perm[eqvt]
section {* Strong Induction Principles*}
(*
Old way of establishing strong induction
principles by chosing a fresh name.
*)
lemma
fixes c::"'a::fs"
assumes a1: "\<And>name c. P c (Var name)"
and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
and a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
shows "P c lam"
proof -
have "\<And>p. P c (p \<bullet> lam)"
apply(induct lam arbitrary: c rule: lam.induct)
apply(perm_simp)
apply(rule a1)
apply(perm_simp)
apply(rule a2)
apply(assumption)
apply(assumption)
apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")
defer
apply(simp add: fresh_def)
apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)
apply(simp add: supp_Pair finite_supp)
apply(blast)
apply(erule exE)
apply(rule_tac t="p \<bullet> Lam name lam" and
s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)
apply(simp del: lam.perm)
apply(subst lam.perm)
apply(subst (2) lam.perm)
apply(rule flip_fresh_fresh)
apply(simp add: fresh_def)
apply(simp only: supp_fn')
apply(simp)
apply(simp add: fresh_Pair)
apply(simp)
apply(rule a3)
apply(simp add: fresh_Pair)
apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
apply(simp)
done
then have "P c (0 \<bullet> lam)" by blast
then show "P c lam" by simp
qed
(*
New way of establishing strong induction
principles by using a appropriate permutation.
*)
lemma
fixes c::"'a::fs"
assumes a1: "\<And>name c. P c (Var name)"
and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
and a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
shows "P c lam"
proof -
have "\<And>p. P c (p \<bullet> lam)"
apply(induct lam arbitrary: c rule: lam.induct)
apply(perm_simp)
apply(rule a1)
apply(perm_simp)
apply(rule a2)
apply(assumption)
apply(assumption)
apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")
apply(erule exE)
apply(rule_tac t="p \<bullet> Lam name lam" and
s="q \<bullet> p \<bullet> Lam name lam" in subst)
defer
apply(simp)
apply(rule a3)
apply(simp add: eqvts fresh_star_def)
apply(drule_tac x="q + p" in meta_spec)
apply(simp)
apply(rule at_set_avoiding2)
apply(simp add: finite_supp)
apply(simp add: finite_supp)
apply(simp add: finite_supp)
apply(perm_simp)
apply(simp add: fresh_star_def fresh_def supp_fn')
apply(rule supp_perm_eq)
apply(simp)
done
then have "P c (0 \<bullet> lam)" by blast
then show "P c lam" by simp
qed
section {* Typing *}
nominal_datatype ty =
TVar string
| TFun ty ty
notation
TFun ("_ \<rightarrow> _")
declare ty.perm[eqvt]
inductive
valid :: "(name \<times> ty) list \<Rightarrow> bool"
where
"valid []"
| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
inductive
typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
where
t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
| t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
| t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"
equivariance valid
equivariance typing
thm valid.eqvt
thm typing.eqvt
thm eqvts
thm eqvts_raw
thm typing.induct[of "\<Gamma>" "t" "T", no_vars]
lemma
fixes c::"'a::fs"
assumes a: "\<Gamma> \<turnstile> t : T"
and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk>
\<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>atom x \<sharp> c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk>
\<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
shows "P c \<Gamma> t T"
proof -
from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
proof (induct)
case (t_Var \<Gamma> x T p c)
then show ?case
apply -
apply(perm_strict_simp)
apply(rule a1)
apply(drule_tac p="p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(rotate_tac 1)
apply(drule_tac p="p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
done
next
case (t_App \<Gamma> t1 T1 T2 t2 p c)
then show ?case
apply -
apply(perm_strict_simp)
apply(rule a2)
apply(drule_tac p="p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(assumption)
apply(rotate_tac 2)
apply(drule_tac p="p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(assumption)
done
next
case (t_Lam x \<Gamma> T1 t T2 p c)
then show ?case
apply -
apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and>
supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
apply(erule exE)
apply(rule_tac t="p \<bullet> \<Gamma>" and s="(q + p) \<bullet> \<Gamma>" in subst)
apply(simp only: permute_plus)
apply(rule supp_perm_eq)
apply(simp add: supp_Pair fresh_star_union)
apply(rule_tac t="p \<bullet> Lam x t" and s="(q + p) \<bullet> Lam x t" in subst)
apply(simp only: permute_plus)
apply(rule supp_perm_eq)
apply(simp add: supp_Pair fresh_star_union)
apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
apply(simp only: permute_plus)
apply(rule supp_perm_eq)
apply(simp add: supp_Pair fresh_star_union)
apply(simp (no_asm) only: eqvts)
apply(rule a3)
apply(simp only: eqvts permute_plus)
apply(simp add: fresh_star_def)
apply(drule_tac p="q + p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(rotate_tac 1)
apply(drule_tac p="q + p" in permute_boolI)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(drule_tac x="d" in meta_spec)
apply(drule_tac x="q + p" in meta_spec)
apply(perm_strict_simp add: permute_minus_cancel)
apply(assumption)
apply(rule at_set_avoiding2)
apply(simp add: finite_supp)
apply(simp add: finite_supp)
apply(simp add: finite_supp)
apply(rule_tac p="-p" in permute_boolE)
apply(perm_strict_simp add: permute_minus_cancel)
(* supplied by the user *)
apply(simp add: fresh_star_prod)
apply(simp add: fresh_star_def)
sorry
qed
then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
then show "P c \<Gamma> t T" by simp
qed
inductive
tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
where
aa: "tt r a a"
| bb: "tt r a b ==> tt r a c"
(* PROBLEM: derived eqvt-theorem does not conform with [eqvt]
equivariance tt
*)
inductive
alpha_lam_raw'
where
a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"
| a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>
alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"
| a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>
alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"
equivariance alpha_lam_raw'
thm eqvts_raw
section {* size function *}
lemma size_eqvt_raw:
fixes t::"lam_raw"
shows "size (pi \<bullet> t) = size t"
apply (induct rule: lam_raw.inducts)
apply simp_all
done
instantiation lam :: size
begin
quotient_definition
"size_lam :: lam \<Rightarrow> nat"
is
"size :: lam_raw \<Rightarrow> nat"
lemma size_rsp:
"alpha_lam_raw x y \<Longrightarrow> size x = size y"
apply (induct rule: alpha_lam_raw.inducts)
apply (simp_all only: lam_raw.size)
apply (simp_all only: alphas)
apply clarify
apply (simp_all only: size_eqvt_raw)
done
lemma [quot_respect]:
"(alpha_lam_raw ===> op =) size size"
by (simp_all add: size_rsp)
lemma [quot_preserve]:
"(rep_lam ---> id) size = size"
by (simp_all add: size_lam_def)
instance
by default
end
lemmas size_lam[simp] =
lam_raw.size(4)[quot_lifted]
lam_raw.size(5)[quot_lifted]
lam_raw.size(6)[quot_lifted]
(* is this needed? *)
lemma [measure_function]:
"is_measure (size::lam\<Rightarrow>nat)"
by (rule is_measure_trivial)
section {* Matching *}
definition
MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"
where
"MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"
(*
lemma MATCH_eqvt:
shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"
unfolding MATCH_def
apply(perm_simp the_eqvt)
apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
apply(simp)
thm eqvts_raw
apply(subst if_eqvt)
apply(subst ex1_eqvt)
apply(subst permute_fun_def)
apply(subst ex_eqvt)
apply(subst permute_fun_def)
apply(subst eq_eqvt)
apply(subst permute_fun_app_eq[where f="M"])
apply(simp only: permute_minus_cancel)
apply(subst permute_prod.simps)
apply(subst permute_prod.simps)
apply(simp only: permute_minus_cancel)
apply(simp only: permute_bool_def)
apply(simp)
apply(subst ex1_eqvt)
apply(subst permute_fun_def)
apply(subst ex_eqvt)
apply(subst permute_fun_def)
apply(subst eq_eqvt)
apply(simp only: eqvts)
apply(simp)
apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")
apply(drule sym)
apply(simp)
apply(rule impI)
apply(simp add: perm_bool)
apply(rule trans)
apply(rule pt_the_eqvt[OF pta at])
apply(assumption)
apply(simp add: pt_ex_eqvt[OF pt at])
apply(simp add: pt_eq_eqvt[OF ptb at])
apply(rule cheat)
apply(rule trans)
apply(rule pt_ex1_eqvt)
apply(rule pta)
apply(rule at)
apply(simp add: pt_ex_eqvt[OF pt at])
apply(simp add: pt_eq_eqvt[OF ptb at])
apply(subst pt_pi_rev[OF pta at])
apply(subst pt_fun_app_eq[OF pt at])
apply(subst pt_pi_rev[OF pt at])
apply(simp)
done
lemma MATCH_cng:
assumes a: "M1 = M2" "d1 = d2"
shows "MATCH M1 d1 x = MATCH M2 d2 x"
using a by simp
lemma MATCH_eq:
assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"
shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"
using a
unfolding MATCH_def
apply(subst if_P)
apply(rule_tac a="r x" in ex1I)
apply(rule_tac x="x" in exI)
apply(blast)
apply(erule exE)
apply(drule_tac x="q" in meta_spec)
apply(auto)[1]
apply(rule the_equality)
apply(blast)
apply(erule exE)
apply(drule_tac x="q" in meta_spec)
apply(auto)[1]
done
lemma MATCH_eq2:
assumes a: "t = l x1 x2" "G x1 x2" "\<And>x1' x2'. t = l x1' x2' \<Longrightarrow> G x1' x2' \<Longrightarrow> r x1' x2' = r x1 x2"
shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = r x1 x2"
sorry
lemma MATCH_neq:
assumes a: "\<And>x. t = l x \<Longrightarrow> G x \<Longrightarrow> False"
shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = d"
using a
unfolding MATCH_def
apply(subst if_not_P)
apply(blast)
apply(rule refl)
done
lemma MATCH_neq2:
assumes a: "\<And>x1 x2. t = l x1 x2 \<Longrightarrow> G x1 x2 \<Longrightarrow> False"
shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = d"
using a
unfolding MATCH_def
apply(subst if_not_P)
apply(auto)
done
*)
ML {*
fun mk_avoids ctxt params name set =
let
val (_, ctxt') = ProofContext.add_fixes
(map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
fun mk s =
let
val t = Syntax.read_term ctxt' s;
val t' = list_abs_free (params, t) |>
funpow (length params) (fn Abs (_, _, t) => t)
in (t', HOLogic.dest_setT (fastype_of t)) end
handle TERM _ =>
error ("Expression " ^ quote s ^ " to be avoided in case " ^
quote name ^ " is not a set type");
fun add_set p [] = [p]
| add_set (t, T) ((u, U) :: ps) =
if T = U then
let val S = HOLogic.mk_setT T
in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
end
else (u, U) :: add_set (t, T) ps
in
(mk #> add_set) set
end;
*}
ML {*
writeln (commas (map (Syntax.string_of_term @{context} o fst)
(mk_avoids @{context} [] "t_Var" "{x}" [])))
*}
ML {*
fun prove_strong_ind (pred_name, avoids) ctxt =
Proof.theorem NONE (K I) [] ctxt
local structure P = Parse and K = Keyword in
val _ =
Outer_Syntax.local_theory_to_proof "nominal_inductive"
"proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
(P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
(P.$$$ ":" |-- P.and_list1 P.term))) []) >> prove_strong_ind)
end;
*}
(*
nominal_inductive typing
*)
(* Substitution *)
definition new where
"new s = (THE x. \<forall>a \<in> s. atom x \<noteq> a)"
primrec match_Var_raw where
"match_Var_raw (Var_raw x) = Some x"
| "match_Var_raw (App_raw x y) = None"
| "match_Var_raw (Lam_raw n t) = None"
quotient_definition
"match_Var :: lam \<Rightarrow> name option"
is match_Var_raw
lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
apply rule
apply (induct_tac a b rule: alpha_lam_raw.induct)
apply simp_all
done
lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
primrec match_App_raw where
"match_App_raw (Var_raw x) = None"
| "match_App_raw (App_raw x y) = Some (x, y)"
| "match_App_raw (Lam_raw n t) = None"
quotient_definition
"match_App :: lam \<Rightarrow> (lam \<times> lam) option"
is match_App_raw
lemma [quot_respect]:
"(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
apply (intro fun_relI)
apply (induct_tac a b rule: alpha_lam_raw.induct)
apply simp_all
done
lemmas match_App_simps = match_App_raw.simps[quot_lifted]
primrec match_Lam_raw where
"match_Lam_raw (S :: atom set) (Var_raw x) = None"
| "match_Lam_raw S (App_raw x y) = None"
| "match_Lam_raw S (Lam_raw n t) = (let z = new (S \<union> (fv_lam_raw t - {atom n})) in Some (z, (n \<leftrightarrow> z) \<bullet> t))"
quotient_definition
"match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
is match_Lam_raw
lemma swap_fresh:
assumes a: "fv_lam_raw t \<sharp>* p"
shows "alpha_lam_raw (p \<bullet> t) t"
using a apply (induct t)
apply (simp add: supp_at_base fresh_star_def)
apply (rule alpha_lam_raw.intros)
apply (metis Rep_name_inverse atom_eqvt atom_name_def fresh_perm)
apply (simp)
apply (simp only: fresh_star_union)
apply clarify
apply (rule alpha_lam_raw.intros)
apply simp
apply simp
apply simp
apply (rule alpha_lam_raw.intros)
sorry
lemma [quot_respect]:
"(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
proof (intro fun_relI, clarify)
fix S t s
assume a: "alpha_lam_raw t s"
show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
using a proof (induct t s rule: alpha_lam_raw.induct)
case goal1 show ?case by simp
next
case goal2 show ?case by simp
next
case (goal3 x t y s)
then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
(match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
then have
c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
e: "alpha_lam_raw (p \<bullet> t) s" and
f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
show ?case
unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
proof
show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
next
have "atom y \<sharp> p" sorry
have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" using swap_fresh by auto
then have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
qed
qed
qed
lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
by (induct x rule: lam.induct) (simp_all add: match_App_simps)
lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
apply (induct x rule: lam.induct)
apply (simp_all add: match_Lam_simps)
apply (simp add: Let_def)
apply (erule conjE)
apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
apply (rule conjI)
apply (simp add: lam.eq_iff)
apply (rule_tac x="(name \<leftrightarrow> z)" in exI)
apply (simp add: alphas)
apply (simp add: eqvts)
apply (simp only: lam.fv(3)[symmetric])
apply (subgoal_tac "Lam name lam = Lam z s")
apply (simp del: lam.fv)
prefer 3
apply (thin_tac "(name \<leftrightarrow> new (S \<union> (fv_lam lam - {atom name}))) \<bullet> lam = s")
apply (simp only: new_def)
apply (subgoal_tac "\<forall>a \<in> S. atom z \<noteq> a")
apply (simp only: fresh_def)
(*thm supp_finite_atom_fset*)
sorry
function subst where
"subst v s t = (
case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
case match_Lam (supp (v,s)) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
by pat_completeness auto
termination apply (relation "measure (\<lambda>(_, _, t). size t)")
apply auto[1]
apply (case_tac a) apply simp
apply (frule lam_some) apply simp
apply (case_tac a) apply simp
apply (frule app_some) apply simp
apply (case_tac a) apply simp
apply (frule app_some) apply simp
done
lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
lemma subst_eqvt:
"p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
proof (induct v s t rule: subst.induct)
case (1 v s t)
show ?case proof (cases t rule: lam_exhaust)
fix n
assume "t = Var n"
then show ?thesis by (simp add: match_Var_simps)
next
fix l r
assume "t = App l r"
then show ?thesis
apply (simp only:)
apply (subst subst.simps)
apply (subst match_Var_simps)
apply (simp only: option.cases)
apply (subst match_App_simps)
apply (simp only: option.cases)
apply (simp only: prod.cases)
apply (simp only: lam.perm)
apply (subst (3) subst.simps)
apply (subst match_Var_simps)
apply (simp only: option.cases)
apply (subst match_App_simps)
apply (simp only: option.cases)
apply (simp only: prod.cases)
apply (subst 1(2)[of "(l, r)" "l" "r"])
apply (simp add: match_Var_simps)
apply (simp add: match_App_simps)
apply (rule refl)
apply (subst 1(3)[of "(l, r)" "l" "r"])
apply (simp add: match_Var_simps)
apply (simp add: match_App_simps)
apply (rule refl)
apply (rule refl)
done
next
fix n t'
assume "t = Lam n t'"
then show ?thesis
apply (simp only: )
apply (simp only: lam.perm)
apply (subst subst.simps)
apply (subst match_Var_simps)
apply (simp only: option.cases)
apply (subst match_App_simps)
apply (simp only: option.cases)
apply (subst match_Lam_simps)
apply (simp only: Let_def)
apply (simp only: option.cases)
apply (simp only: prod.cases)
apply (subst (2) subst.simps)
apply (subst match_Var_simps)
apply (simp only: option.cases)
apply (subst match_App_simps)
apply (simp only: option.cases)
apply (subst match_Lam_simps)
apply (simp only: Let_def)
apply (simp only: option.cases)
apply (simp only: prod.cases)
apply (simp only: lam.perm)
apply (simp only: lam.eq_iff)
sorry
qed
qed
lemma subst_proper_eqs:
"subst y s (Var x) = (if x = y then s else (Var x))"
"subst y s (App l r) = App (subst y s l) (subst y s r)"
"atom x \<sharp> (t, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"
apply (subst subst.simps)
apply (simp only: match_Var_simps)
apply (simp only: option.simps)
apply (subst subst.simps)
apply (simp only: match_App_simps)
apply (simp only: option.simps)
apply (simp only: prod.simps)
apply (simp only: match_Var_simps)
apply (simp only: option.simps)
apply (subst subst.simps)
apply (simp only: match_Lam_simps)
apply (simp only: match_Var_simps)
apply (simp only: match_App_simps)
apply (simp only: option.simps)
apply (simp only: Let_def)
apply (simp only: option.simps)
apply (simp only: prod.simps)
sorry
end