How to define a semilattice of lists? - isabelle

Question

I'm trying to define an upper semilattice:
A ≺ B
... ≺ C [B,B,B] ≺ C [B,B] ≺ C [B] ≺ B
C [A] ≺ C [B]
C [A,A] ≺ C [A,B] ≺ C [B,B]
C [A,A] ≺ C [B,A] ≺ C [B,B]
C [A,A,A] ≺ C [A,A,B] ≺ C [A,B,B] ≺ C [B,B,B]
C [A,A,A] ≺ C [A,B,A] ≺ C [A,B,B] ≺ C [B,B,B]
C [A,A,A] ≺ C [A,A,B] ≺ C [B,A,B] ≺ C [B,B,B]
C [A,A,A] ≺ C [B,A,A] ≺ C [B,A,B] ≺ C [B,B,B]
C [A,A,A] ≺ C [A,B,A] ≺ C [B,B,A] ≺ C [B,B,B]
C [A,A,A] ≺ C [B,A,A] ≺ C [B,B,A] ≺ C [B,B,B]
and so on...
I need to prohibit a direct relations of the form:
C [A,A] ≺ C [B,B]
C [A,A,B] ≺ C [B,B,B]
and so on...
Only one element must differ in preceding elements. I will define this indirect relations as a transitive closure.
I tried to define it as follows:
datatype t = A | B | C "t list"
definition "only_one p xs ys ≡
let xys = zip xs ys in
length xs = length ys ∧
list_all (λ(x, y). x = y ∨ p x y) xys ∧
length xs =
length (takeWhile (λ(x, y). x = y) xys) +
length (takeWhile (λ(x, y). x = y) (rev xys)) + 1"
inductive prec_t ("_ ≺ _" [65, 65] 65) where
"A ≺ B"
| "C [B] ≺ B"
| "C (xs#[B]) ≺ C xs"
| "only_one (λx y. x ≺ y) xs ys ⟹
C xs ≺ C ys"
But I get the following error:
Proof failed.
1. ⋀x y xa xb xs ys.
x (?x47 x y xa xb xs ys) (?x48 x y xa xb xs ys) ⟶
y (?x47 x y xa xb xs ys) (?x48 x y xa xb xs ys) ⟹
only_one x xs ys ⟶ only_one y xs ys
The error(s) above occurred for the goal statement⌂:
mono
(λp x1 x2.
x1 = A ∧ x2 = B ∨
x1 = C [B] ∧ x2 = B ∨
(∃xs. x1 = C (xs # [B]) ∧ x2 = C xs) ∨ (∃xs ys. x1 = C xs ∧ x2 = C ys ∧ only_one p xs ys))
Could you suggest how to fix it?
UPDATE
I redefined the condition checker as follows. But it doesn't help.
primrec only_one' :: "bool ⇒ ('a ⇒ 'a ⇒ bool) ⇒ 'a list ⇒ 'a list ⇒ bool" where
"only_one' found p xs [] = (case xs of [] ⇒ found | _ ⇒ False)"
| "only_one' found p xs (y # ys) = (case xs of [] ⇒ False | z # zs ⇒
if z = y then only_one' found p zs ys else
let found' = p z y in
if found ∧ found' then False else only_one' found' p zs ys)"
abbreviation "only_one ≡ only_one' False"
UPDATE 2
An inductive definition doesn't help too:
inductive only_one :: "bool ⇒ ('a ⇒ 'a ⇒ bool) ⇒ 'a list ⇒ 'a list ⇒ bool" where
"only_one True p [] []"
| "x = y ⟹
only_one found p xs ys ⟹
only_one found p (x#xs) (y#ys)"
| "p x y ⟹
found = False ⟹
only_one True p xs ys ⟹
only_one found p (x#xs) (y#ys)"

Answer 1

Your inductive definition is not accepted, since Isabelle does not know that only_one is monotone.
Therefore, the following code in front of your inductive definition should solve your problem.
lemma only_one_mono: "(⋀ x y. x ∈ set xs ⟹ y ∈ set ys ⟹ p x y ⟶ q x y) ⟹
only_one p xs ys ⟶ only_one q xs ys" sorry
declare only_one_mono[mono]
Best, René