# Re: [isabelle] Congruence rule for Let

Hi Andreas,

using the function package, I'd like to define a function whose definition contains a number of Let expressions. In the generated induction rule, a term "Let t (%x. y)" yields the induction hypothesis "!!x. x = t ==> P (y x)" However, I would like to get "P (y t)" directly. How do I have to change the congruence rule for Let to achieve this?

I tried two alternatives with fundef_cong:
- "[| M = N; f N = g N |] ==> Let M f = Let N g" raises an exception:

*** exception THM 1 raised (line 421 of "drule.ML"): COMP
*** At command "function".

- "[| M = N; f M = g N |] ==> Let M f = Let N g" eliminates the quantifier, but produces far to many induction hypotheses.

What is the right congruence rule for this?

Actually, I had expected that the first rule works. I need to dig into this again. The rule that should definitely work is the following

"f M = g N ==> Let M f = Let N g"

It has the same effect on the induction rule as unfolding all lets. Can you try if it works for your function?

Jeremy Dawson wrote:
This seems easy enough using the functions in ch 5 of the Reference Manual

val iax = "!!x. x = t ==> P (y x)" : Thm.thm
val ax' = forall_elim_var 0 iax ;
val ax' = "?x = t ==> P (y ?x)" : Thm.thm
refl RS ax' ;
val it = "P (y t)" : Thm.thm
The ind.hyp. Andreas is referring to appears nested inside the induction rule, where it cannot be easily manipulated by hand...

Alex

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