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Require Import Equality.
Parameter Patch : nat -> nat -> Set.
Inductive Catch (from to : nat) : Type
:= MkCatch : forall (p : Patch from to),
Catch from to.
Implicit Arguments MkCatch [from to].
Inductive CatchCommute5
: forall {from mid1 mid2 to : nat},
Catch from mid1
-> Catch mid1 to
-> Catch from mid2
-> Catch mid2 to
-> Prop
:= MkCatchCommute5 :
forall {from mid1 mid2 to : nat}
(p : Patch from mid1)
(q : Patch mid1 to)
(q' : Patch from mid2)
(p' : Patch mid2 to),
CatchCommute5 (MkCatch p) (MkCatch q) (MkCatch q') (MkCatch p').
Inductive CatchCommute {from mid1 mid2 to : nat}
(p : Catch from mid1)
(q : Catch mid1 to)
(q' : Catch from mid2)
(p' : Catch mid2 to)
: Prop
:= isCatchCommute5 : forall (catchCommuteDetails : CatchCommute5 p q q' p'),
CatchCommute p q q' p'.
Notation "<< p , q >> <~> << q' , p' >>"
:= (CatchCommute p q q' p')
(at level 60, no associativity).
Lemma CatchCommuteUnique2 :
forall {from mid mid' to : nat}
{p : Catch from mid} {q : Catch mid to}
{q' : Catch from mid'} {p' : Catch mid' to}
{q'' : Catch from mid'} {p'' : Catch mid' to}
(commute1 : <<p, q>> <~> <<q', p'>>)
(commute2 : <<p, q>> <~> <<q'', p''>>),
(p' = p'') /\ (q' = q'').
Proof with auto.
intros.
set (X := commute2).
dependent destruction commute1;
dependent destruction catchCommuteDetails;
dependent destruction commute2;
dependent destruction catchCommuteDetails generalizing X.
Admitted.
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