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From Stdlib Require Import Lia.
From Stdlib Require Import ZArith.
Import ZifyClasses.
Module Test1.
Record Z2@{u} : Type@{u} := MkZ2 { unZ2 : Z }.
Global Instance Inj_Z2_Z : InjTyp Z2 Z :=
{ inj := unZ2
; pred := fun _ => True
; cstr := fun _ => I
}.
Add Zify InjTyp Inj_Z2_Z.
Lemma eq_Z2_inj :
forall (n m : Z2),
n = m <-> unZ2 n = unZ2 m.
Proof.
Admitted.
Global Instance Op_eq : BinRel (@eq Z2) :=
{ TR := @eq Z
; TRInj := eq_Z2_inj
}.
Add Zify BinRel Op_eq.
Theorem lia_refl_ex : forall (a b : Z2), a = a.
Proof.
lia.
Defined.
Fail Constraint mkrel.u0 < unZ2.u.
End Test1.
Module Test2.
(* we need a separate copy of Z2 to test a different case because
otherwise the constraint is set in one of the tests *)
Record Z2@{u} : Type@{u} := MkZ2 { unZ2 : Z }.
Global Instance Inj_Z2_Z : InjTyp Z2 Z :=
{ inj := unZ2
; pred := fun _ => True
; cstr := fun _ => I
}.
Add Zify InjTyp Inj_Z2_Z.
Lemma eq_Z2_inj :
forall (n m : Z2),
n = m <-> unZ2 n = unZ2 m.
Proof.
Admitted.
Global Instance Op_eq : BinRel (@eq Z2) :=
{ TR := @eq Z
; TRInj := eq_Z2_inj
}.
Add Zify BinRel Op_eq.
Theorem lia_refl_ex : forall (a b : Z2), a = b -> True.
Proof.
zify.
exact I.
Defined.
Fail Constraint mkrel.u0 < Z2.u.
End Test2.
Module Test3.
Record Z2@{u} : Type@{u} := MkZ2 { unZ2 : Z }.
Global Instance Inj_Z2_Z : InjTyp Z2 Z :=
{ inj := unZ2
; pred := fun _ => True
; cstr := fun _ => I
}.
Add Zify InjTyp Inj_Z2_Z.
Lemma eq_Z2_inj :
forall (n m : Z2),
n = m <-> unZ2 n = unZ2 m.
Proof.
Admitted.
Global Instance Op_eq : BinRel (@eq Z2) :=
{ TR := @eq Z
; TRInj := eq_Z2_inj
}.
Add Zify BinRel Op_eq.
Constraint mkrel.u0 < Z2.u.
Theorem lia_refl_ex : forall (a b : Z2), a = a.
Proof.
Fail lia.
Abort.
End Test3.
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