# [isabelle-dev] simplifier and subgoaler not transitive

Thomas Sewell Thomas.Sewell at nicta.com.au
Wed May 23 03:39:53 CEST 2012

```Question: it looks to me like "(atom v # (x, y)) = (atom v # x & atom v
# y)"

It also looks like what you're trying to do is allow the system to
reason with the above equality without actually giving it that equality.
It looks like you've provided the equality in one direction as a rewrite
rule and in the other direction by adjusting mksimps (just guessing
here, but that's what it looks like).

Did I guess right? If so, I know why that won't work :-) The new rewrite
rules created by mksimps aren't themselves candidates for
simplification, so the system won't apply Nominal2_Base.fresh_at_base_2
to them, which was what resulted in further progress on goal #2.

Those are all giant guesses. Am I anywhere near the mark?

More directly, why not just add the rewrite at the top of this email to
the simpset? This will reduce all of these sharp statements to trivial
inequalities. This is the approach that fits with the general design of
the simplifier. Not the structure you want? Too many inequalities? In
that case you really need a guided solver - giving the simplifier
opportunities for wild exploration will just slow everything down.

Yours,
Thomas.

On 23/05/12 02:23, Christian Urban wrote:
> Dear All,
>
> Assuming that this is about the bowels of the simplifier,
> I hope this is the right place to ask the following question.
>
> The simplifier has a subgoaler, which helps with rewriting
> conditional lemmas. This simplifiying/subgoaling process seems
> to be not transitive (probably is not meant to be). The problem
> that arises for me is the following: I have set up the simplifier
> to automatically solve the first two lemmas:
>
>    lemma "atom v # (x1, x2) ==>  atom v # x1"
>    apply(simp)
>    done
>
>    lemma "atom v # x1 ==>  v \<noteq>  x1"
>    apply(simp)
>    done
>
> but it chokes, if I am trying to string both lemmas
> together
>
>    lemma "atom v # (x1, x2) ==>  v \<noteq>  x1"
>    apply(simp) --"fails"
>
> Is there some magic that I can make the simplifier to
> deal also with the latter goal?
>
> The cool thing about jEdit is that I have the simplifier
> traces of all three goals next to each other (the trick
> is to disable the auto update). Unfortunately, I am
> not very good at reading these traces. The only thing I
> can say is that the simplifier starts off with goal 1
> and 3 in the same direction, but then things start to
> diverge. Is there a place where one can read up about
> the tracing information of the simplifier? The traces
> are attached for reference.
>
> Best wishes and thanks for any help,
> Christian
>
>
>
> GOAL 1
> ======
>
>   [1]SIMPLIFIER INVOKED ON THE FOLLOWING TERM:
> atom v ♯ (x1, x2) ⟹ atom v ♯ x1
>   [1]Applying instance of rewrite rule "??.unknown":
> ?a1 ♯ ?x1.1 ⟹ ?a1 ♯ ?x2.1 ⟹ ?a1 ♯ (?x1.1, ?x2.1) ≡ True
>   [1]Trying to rewrite:
> atom v ♯ x1 ⟹ atom v ♯ t ⟹ atom v ♯ (x1, x2) ≡ True
>   [2]SIMPLIFIER INVOKED ON THE FOLLOWING TERM:
> atom v ♯ x1
>   [1]FAILED
> atom v ♯ x1 ⟹ atom v ♯ t ⟹ atom v ♯ (x1, x2) ≡ True
> atom v ♯ x1 ≡ True
> atom v ♯ t ≡ True
>   [1]Applying instance of rewrite rule "??.unknown":
> atom v ♯ x1 ≡ True
>   [1]Rewriting:
> atom v ♯ x1 ≡ True
>   proof (prove): step 1
>
> goal:
> No subgoals!
>
>
> GOAL 2
> ======
>
>   [1]SIMPLIFIER INVOKED ON THE FOLLOWING TERM:
> atom v ♯ x1 ⟹ v ≠ x1
>   [1]Applying instance of rewrite rule "Nominal2_Base.fresh_at_base_2":
> ?a1 ♯ ?b1 ≡ ?a1 ≠ atom ?b1
>   [1]Rewriting:
> atom v ♯ x1 ≡ atom v ≠ atom x1
>   [1]Applying instance of rewrite rule "Nominal2_Base.at_base_class.atom_eq_iff":
> atom ?a1 = atom ?b1 ≡ ?a1 = ?b1
>   [1]Rewriting:
> atom v = atom x1 ≡ v = x1
> v = x1 ≡ False
>   [1]Applying instance of rewrite rule "??.unknown":
> v = x1 ≡ False
>   [1]Rewriting:
> v = x1 ≡ False
>   [1]Applying instance of rewrite rule "HOL.simp_thms_8":
> ¬ False ≡ True
>   [1]Rewriting:
> ¬ False ≡ True
>   proof (prove): step 1
>
> goal:
> No subgoals!
>
>
> Goal 3
> ======
>
>   [1]SIMPLIFIER INVOKED ON THE FOLLOWING TERM:
> atom v ♯ (x1, x2) ⟹ v ≠ x1
>   [1]Applying instance of rewrite rule "??.unknown":
> ?a1 ♯ ?x1.1 ⟹ ?a1 ♯ ?x2.1 ⟹ ?a1 ♯ (?x1.1, ?x2.1) ≡ True
>   [1]Trying to rewrite:
> atom v ♯ x1 ⟹ atom v ♯ t ⟹ atom v ♯ (x1, x2) ≡ True
>   [2]SIMPLIFIER INVOKED ON THE FOLLOWING TERM:
> atom v ♯ x1
>   [2]Applying instance of rewrite rule "Nominal2_Base.fresh_at_base_2":
> ?a1 ♯ ?b1 ≡ ?a1 ≠ atom ?b1
>   [2]Rewriting:
> atom v ♯ x1 ≡ atom v ≠ atom x1
>   [2]Applying instance of rewrite rule "Nominal2_Base.at_base_class.atom_eq_iff":
> atom ?a1 = atom ?b1 ≡ ?a1 = ?b1
>   [2]Rewriting:
> atom v = atom x1 ≡ v = x1
>   [1]FAILED
> atom v ♯ x1 ⟹ atom v ♯ t ⟹ atom v ♯ (x1, x2) ≡ True
> atom v ♯ x1 ≡ True