Nominal Inversion Principles

4 Stefan Berghofer and Christian Urban
8 x s 2 s 1: u 1 = App (Lam x:s 1) s 2 ^ u 2 = s 1[x:=s 2] ) P
8 s 1 s 2 t: u 1 = App s 1 t ^ u 2 = App s 2 t ^ s 1 ! s 2 ) P
8 s 1 s 2 t: u 1 = App t s 1 ^ u 2 = App t s 2 ^ s 1 ! s 2 ) P
8 s 1 s 2 x: u 1 = Lam x:s 1 ^ u 2 = Lam x:s 2 ^ s 1 ! s 2 ) P
u 1 ! u 2 ) P (4)
8 x T: ¡ = ^ u = Var x ^ U = T ^ valid ^ (x; T) 2 ) P
8 t 1 T 1 T 2 t 2: ¡ = ^ u = App t 1 t 2 ^ U = T 2 ^ ` t 1 : T 1 ! T 2 ^ ` t 2 : T 1 ) P
8 x T 1 t T 2: ¡ = ^ u = Lam x:t ^ U = T 1 ! T 2 ^ (x; T 1):: ` t : T 2 ) P
¡ ` u : U ) P (5)
Fig. 1. Inversion principles derived by Isabelle/HOL for the inductive predicates beta-reduction
and typing.
If we use inversion principle for ! (i.e. (4)) and invert Var x ! u 0 , we obtain the
following four subgoals:
8 x 0 s 2 s 1 : Var x = App (Lam x 0 :s 1 ) s 2 ^ u 0 = s 1 [x 0 :=s 2 ] ^ : : : ) ` u 0 : T
8 s 1 s 2 t: Var x = App s 1 t ^ u 0 = App s 2 t ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
8 s 1 s 2 t: Var x = App t s 1 ^ u 0 = App t s 2 ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
8 s 1 s 2 x 0 : Var x = Lam x 0 :s 1 ^ u 0 = Lam x 0 :s 2 ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
The left-hand sides of these subgoals all reduce to False because the term constructors
are in conflict (Var can never be equal to App). Therefore we can quickly, like in the
informal reasoning, discharge all subgoals.
In case (ii) where we invert App t 1 t 2 ! u 0 , we obtain the following four subgoals:
8 x s 2 s 1 : App t 1 t 2 = App (Lam x:s 1 ) s 2 ^ u 0 = s 1 [x:=s 2 ] ^ : : : ) ` u 0 : T
8 s 1 s 2 t: App t 1 t 2 = App s 1 t ^ u 0 = App s 2 t ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
8 s 1 s 2 t: App t 1 t 2 = App t s 1 ^ u 0 = App t s 2 ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
8 s 1 s 2 x: App t 1 t 2 = Lam x:s 1 ^ u 0 = Lam x:s 2 ^ s 1 ! s 2 ^ : : : ) ` u 0 : T
The fourth subgoal can again be discharged because of the conflicting equality between
App and Lam. The reasoning in the second and third is very similar with the informal
inversion by matching, because the App-term constructor is injective and therefore we
can infer
App t 1 t 2 = App s 1 t ) t 1 = s 1 ^ t 2 = t; and
App t 1 t 2 = App t s 1 ) t 1 = t ^ t 2 = s 1
(6)
which are the same equations we would have got by the informal inversion by matching.
The first subgoal (corresponding to b 1 ) is more complicated: although we obtain by
injectivity of App the equations t 1 = Lam x:s 1 and t 2 = s 2 , we will encounter problems
with inverting the typing judgement ` Lam x:s 1 : T 1 ! T 2 . That is, we will not be
able to infer that (x; T 1 ):: ` s 1 : T 2 holds. This is because Lam is not injective and
we cannot reason as in (6).
We encounter the same problem with the reasoning in case (iii). There we have to
invert the reduction Lam x:t ! u 0 and obtain by using the first inversion principle
from (4) the following four subgoals: