Email Gateway (Aug 19 2022 at 08:41): Email Gateway (Aug 19 2022 at 08:41):From: Holger Blasum <hbl@sysgo.com>
Dear all,
this is probably an easy one, I am trying to define the range
of a function that uses pattern matching on a datatype (num_t).
I can show it by "auto" in lemma f_range (see below) on
a function that is num_t-valued. I seem to be missing an intermediate
step once I am wrapping that num_t into a record num_wrapper_t,
(see proof attempt at f_wrapper_def below).
theory record_valued_function
imports Main
begin
datatype num_t = ONE | TWO
(* The following lemma has been proved by "auto". *)
definition f::"num_t => nat set" where
"f n = (if (n = ONE) then {1} else {2})"
lemma f_range: "Union ({x. EX n. x = f n}) = {1,2}"
proof-
from f_def show ?thesis by auto
qed
(* This is the lemma where I am missing an intermediate step. *)
record num_wrapper_t = num::num_t
definition f_wrapper::"num_wrapper_t => nat set" where
"f_wrapper r = (if ((num r) = ONE) then {1} else {2})"
lemma f_wrapper_range: "Union ({x. (EX r. x = f_wrapper r)}) = {1,2}"
proof-
from f_wrapper_def have r1: "ALL r. num r = ONE
--> f_wrapper r = {1}" by fastforce
from f_wrapper_def have r2: "ALL r. num r = TWO
--> f_wrapper r = {2}" by fastforce
from r1 and r2 and f_wrapper_def show ?thesis
sorry <-- fails
Suggestions on how to complete the second lemma would be welcome,
Email Gateway (Aug 19 2022 at 08:41):From: Thomas Sewell <Thomas.Sewell@nicta.com.au>
You can solve this problem via:
lemma f_wrapper_range: "Union ({x. (EX r. x = f_wrapper r)}) = {1,2}"
by (auto simp add: f_wrapper_def num_wrapper_t.splits)
To be slightly clearer:
lemma f_wrapper_range: "Union ({x. (EX r. x = f_wrapper r)}) = {1,2}"
proof -
have exs: "EX x. num x = ONE" "EX x. num x ~= ONE"
by (simp_all add: num_wrapper_t.splits exI[where x=TWO])
thus ?thesis
by (auto simp add: f_wrapper_def)
qed
What you need to show is that there are objects in the num_wrapper_t
type with num taking both values. That's not syntactically clear: if num
x had a more limited range, f_wrapper would too.
Yours,
Thomas.
Email Gateway (Aug 19 2022 at 08:41):From: Tjark Weber <webertj@in.tum.de>
Main already provides image (`) and range operators, cf. Section 4 of
http://www.cl.cam.ac.uk/research/hvg/isabelle/dist/Isabelle2012/doc/main.pdf
So you could simply claim "range f = {1,2}".
Best regards,
Tjark
Email Gateway (Aug 19 2022 at 09:16):From: Holger Blasum <hbl@sysgo.com>
Hi Thomas, Tjark,
I realize I never followed up this on. Yup, worked.
The "splits" lemma also made me curious about what exactly gets
introduced after declaring a datatype and a record,
so I also played a bit with "print_theorems" after declaring
the datatype and the record - useful information.
Thanks,
Last updated: Apr 26 2024 at 20:16 UTC