[isabelle] Set induction and pair datatype

I have the following problem with set induction and pair datatypes,
illustrated in the pice of code below:

-- "Defining transitive closure of labeled transition system as
inductive set"
types ('c,'a) LTS = "('c \<times> 'a \<times> 'c) set"

text {* Transitive closure of LTS *}
consts trcl :: "('c,'a) LTS \<Rightarrow> ('c,'a list) LTS"

inductive "trcl t"
  empty[simp]: "(c,[],c) \<in> trcl t"
  cons[simp]: "\<lbrakk> (c,a,c') \<in> t; (c',w,c'') \<in> trcl t
\<rbrakk> \<Longrightarrow> (c,a#w,c'') \<in> trcl t"

-- "Now do some very simple inductive proof, using trcl.induct"
consts T':: "(nat \<times> bool \<times> nat) set"
consts Prop':: "nat \<Rightarrow> bool"

lemma preserves': "(a,e,a')\<in>T' \<Longrightarrow> Prop' a
\<Longrightarrow> Prop' a'" sorry

lemma preserves_trcl': "(a,w,a')\<in>trcl T' \<Longrightarrow> Prop' a
\<Longrightarrow> Prop' a'"
  apply (induct rule: trcl.induct)
  apply (auto simp add: preserves')
-- "Ok, this works fine !"

-- "Now we want to do the same for states that are pairs: "
consts T:: "((nat \<times> nat) \<times> bool \<times> (nat \<times>
nat)) set"
consts Prop:: "nat \<Rightarrow> nat \<Rightarrow> bool"

lemma preserves: "((a,b),e,(a',b'))\<in>T \<Longrightarrow> Prop a b
\<Longrightarrow> Prop a' b'" sorry

lemma preserves_cl: "((a,b),w,(a',b'))\<in>trcl T \<Longrightarrow> Prop
a b \<Longrightarrow> Prop a' b'"
  apply (induct rule: trcl.induct)
  This did not get the induction correctly, there are the following
    1. \<And>c. Prop a b \<Longrightarrow> Prop a' b'
    2. \<And>aa c c' c'' w. \<lbrakk>(c, aa, c') \<in> T; (c', w, c'')
\<in> trcl T; Prop a b \<Longrightarrow> Prop a' b'; Prop a b\<rbrakk>
\<Longrightarrow> Prop a' b'
  the first one is obviously not provable

What can I do about this ? Currently I'm working around by providing a
specialized induction rule for pairs:
lemma trcl_pair_induct[induct set]:
  "\<lbrakk>((xc1,xc2), xb, (xa1,xa2)) \<in> trcl t; \<And>c1 c2. P c1
c2 [] c1 c2; \<And>a c1 c2 c1' c2' c1'' c2'' w. \<lbrakk>((c1,c2), a,
(c1',c2')) \<in> t; ((c1',c2'), w, (c1'',c2'')) \<in> trcl t; P c1' c2'
w c1'' c2''\<rbrakk> \<Longrightarrow> P c1 c2 (a # w) c1'' c2''\<rbrakk>
  \<Longrightarrow> P xc1 xc2 xb xa1 xa2"
  using trcl.induct[of "(xc1,xc2)" xb "(xa1,xa2)" t "\<lambda>c w c'.
let (c1,c2)=c in let (c1',c2')=c' in P c1 c2 w c1' c2'"] by auto

Is this the way to go, or is there another (nicer) possibility, i.e. one
generic scheme that can handle pairs, triples and so on

Many thanks in advance, yours

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