# Re: [isabelle] monotonicity lemmas for embedded lists

```Peter Sewell wrote:
```
```Dear Isabelle developers,

we have a lot of datatypes with embedded lists, as in the example
below (this is a small example cut down from large (and automatically
generated) examples).  Defining inductive relations over these needs
some additional monotonicity lemmas, here "tmp" and "tmp2".  In
general we need many analogues to tmp2, all of a similar form, e.g.
(from a different example):

```
lemma tmp7: " A <= B ==> ALL x. (%(l_, p_, T_, D_). (p_, T_, D_) : (%y. Inr (Inr (Inr y))) -` A) x - --> (%(l_, p_, T_, D_). (p_, T_, D_) : (%y. Inr (Inr (Inr y))) -` B) x" by blast
```
Dear Peter,

one way to tell the monotonicity prover how to deal with abstractions
over tuples (also known as "split") is to add the following equation
as a monotonicity rule:

lemma [mono]: "split f p = f (fst p) (snd p)" by (simp add: split_def)

This will just unfold the definition of split during the proof of
monotonicity, i.e. the prover should be able to deal with any abstraction
over n-tuples. As far as the inverse image operator -` is concerned,
there already is a monotonicity rule of the form

A <= B ==> f -` A <= f -` B

in the default set of rules used by the monotonicity prover.
However, rather than using the inverse image operator, it seems
easier to define several inductive sets simultaneously.

```
```lemma tmp:"
! x. f x --> g x ==> list_all (%b. b) (map f l_T_list)-->
list_all (%b. b) (map g l_T_list) "
```
```
Note that a meta-level quantifier is required in the premise, i.e.

(!! x. f x --> g x) ==> list_all (%b. b) (map f l_T_list)-->
list_all (%b. b) (map g l_T_list)

See section 2.8.3 "Monotonicity theorems" in the Isabelle/HOL manual
for a description of the different forms of monotonicity rules accepted
by the inductive definition package.

Greetings,
Stefan

--
Dr. Stefan Berghofer               E-Mail: berghofe at in.tum.de
Institut fuer Informatik           Phone: +49 89 289 17328
Technische Universitaet Muenchen   Fax:   +49 89 289 17307
Boltzmannstr. 3                    Room: 01.11.059
85748 Garching, GERMANY            http://www.in.tum.de/~berghofe

```

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