[Coq-Club] Rewriting through coercions

[richard Dapoigny]

Hi,

I have some question about rewriting while some coercions have been set.
Some are working, others not.

Here is a minimal fragment to assess the problem.

Universe Name.

Variable name         : Type@{Name}.
Variable plural       : Type@{Name}.
Variable particular   : Type@{Name}.

Variable c01 : plural -> name. Coercion c01 : plural >-> name.
Variable c02 : particular -> name. Coercion c02 : particular >-> name.


Inductive eps :particular -> name -> Prop :=
 | particularEquality: forall(A:particular)(B:particular), eps A B
 | epsilon : forall(A:particular)(b:plural), eps A b.

Notation "A 'ε' b" := (eps A b)  (at level 40).

Definition Eq_name (a b:name) := @eq name a b.

...

Class el :Type@{Name} := mkEl { el01  : particular }.
Variable  c04 : el -> plural. Coercion c04 : el >-> plural.

Axiom isElement  : forall (elB:el)(A B:particular), A ε elB <-> Eq_name
B (@el01 elB) \/ forall ptB:part, Eq_name (@el01 elB)(@pt01 ptB) /\ A ε ptB.

Class klass : Type@{Name}  := mkKlass {
    k01  : name;
    k02  : exists B:particular, B ε k01 }.

(* Klass sous-type des particulars *)
Variable c07 : klass -> particular. Coercion c07 : klass >-> particular.

Then in order to prove the following theorem :

Lemma Th10 : forall (A:particular)(a:plural)(klA:klass), A ε a ->
Eq_name A (@k01 klA) -> A ε klA.
Proof.
intros A a kA H H'.
rewrite isKlass.
split.
- intros B H1 el1 H2.
  apply (isElement el1 B A).
  left.
  symmetry.
  exact H2.
- intros B elA elB H2.
  destruct H2 as [H2 H3].
  destruct H3 as [H3 H4].
  exists elA, elB.
  split.
    + rewrite <-H'.
     rewrite H2.

===================================================================

the message is: Found no subterm matching "c02 el01" in the current goal.

with hypotheses :

A : particular
a : plural
kA : klass
H : A ε a
H' : Eq_name A k01
B : particular
elA, elB : el
H2 : Eq_name el01 A
H3 : B ε elA
H4 : Eq_name el01 B
______________________________________(1/1)
el01 ε A

Having some idea about this message while c02 el01 has the right type?

Thanks in advance.

[Fabian Kunze]

The minimal example does not compile for me (I assume you leave something
out at "..."? ). I would have had a look at it if it would compile.

Post by richard Dapoigny
Hi,
I have some question about rewriting while some coercions have been set.
Some are working, others not.
Here is a minimal fragment to assess the problem.
Universe Name.
Variable c01 : plural -> name. Coercion c01 : plural >-> name.
Variable c02 : particular -> name. Coercion c02 : particular >-> name.
Inductive eps :particular -> name -> Prop :=
| particularEquality: forall(A:particular)(B:particular), eps A B
| epsilon : forall(A:particular)(b:plural), eps A b.
Notation "A 'ε' b" := (eps A b) (at level 40).
...
Variable c04 : el -> plural. Coercion c04 : el >-> plural.
Axiom isElement : forall (elB:el)(A B:particular), A ε elB <-> Eq_name B
k01 : name;
k02 : exists B:particular, B ε k01 }.
(* Klass sous-type des particulars *)
Variable c07 : klass -> particular. Coercion c07 : klass >-> particular.
Lemma Th10 : forall (A:particular)(a:plural)(klA:klass), A ε a -> Eq_name
Proof.
intros A a kA H H'.
rewrite isKlass.
split.
- intros B H1 el1 H2.
apply (isElement el1 B A).
left.
symmetry.
exact H2.
- intros B elA elB H2.
destruct H2 as [H2 H3].
destruct H3 as [H3 H4].
exists elA, elB.
split.
+ rewrite <-H'.
rewrite H2.
===================================================================
the message is: Found no subterm matching "c02 el01" in the current goal.
A : particular
a : plural
kA : klass
H : A ε a
H' : Eq_name A k01
B : particular
elA, elB : el
H2 : Eq_name el01 A
H3 : B ε elA
H4 : Eq_name el01 B
______________________________________(1/1)
el01 ε A
Having some idea about this message while c02 el01 has the right type?
Thanks in advance.
--
*Richard Dapoigny*
Associate Professor - LISTIC Laboratory
University Savoie Mont-Blanc
5, chemin Bellevue
<https://maps.google.com/?q=5,+chemin+Bellevue&entry=gmail&source=g>
Po. Box 80439
Annecy-le-Vieux 74940
FRANCE
https://www.listic.univ-smb.fr/en/presentation-en/members/lecturers/richard-dapoigny-en/

[richard Dapoigny]

Hi Fabian,

here is the full code. The problem is in lemma Th5.

Require Import Relation_Definitions.
Require Import Coq.Program.Basics.
Require Import Coq.Classes.EquivDec.
Require Import Coq.Program.Tactics.
Require Import Coq.Classes.SetoidClass.
Require Import Coq.Classes.SetoidTactics.
Require Import Coq.Logic.Decidable.
Require Import Unicode.Utf8.
Require Import Coq.Logic.Classical_Prop.

Section Mereo.

Universe Name.

Variable name         : Type@{Name}.
Variable plural       : Type@{Name}.
Variable particular   : Type@{Name}.

Variable c01 : plural -> name. Coercion c01 : plural >-> name.
Variable c02 : particular -> name. Coercion c02 : particular  >-> name.


Inductive eps :particular -> name -> Prop :=
 | particularEquality: forall(A:particular)(B:particular), eps A B
 | epsilon : forall(A:particular)(b:plural), eps A b.

Notation "A 'ε' b" := (eps A b)  (at level 40).

Definition Eq_name (a b:name) := @eq name a b.

Variable Λ : plural.
Variable Ω : plural.

Axiom contra_name : forall (A:particular), A ε Λ <-> False.
Axiom A_is_object : forall (A:particular), A ε Ω <-> True.

Axiom A01 : forall (A:particular), A ε A. (* reflexivity *)
Axiom A02 : forall (A B C:particular), (A ε B /\ C ε B) -> A ε C.
Axiom A03 : forall (A:particular)(a:plural), A ε a <-> forall D, (D ε A
-> D ε a).
Axiom A04 : forall (A:particular), A ε A -> exists b:plural, A ε b.


Lemma Ax01 : forall (A:particular)(a:plural), A ε a -> (exists
B:particular, B ε A).
Proof.
intros.
exists A.
apply A01.
Qed.

Lemma Equiv_Lesniewski_axiom_right : forall (A:particular)(a:plural), A
ε a -> (exists B:particular, B ε A) /\
                                       (forall (C D:particular), (D ε A
/\ C ε A) -> D ε C) /\
                                       forall (D:particular), D ε A ->
D ε a.
Proof.
intros A b H1.
split.
 + apply (Ax01 A b);assumption.
 + split.
    - intros;apply (A02 D A C);assumption.
    - intros B H2;rewrite (A03 A b) in H1;specialize (H1 B);apply H1 in
H2;assumption.
Qed.

Lemma Equiv_Lesniewski_axiom_left : forall (A:particular)(a:plural),
((exists B:particular, B ε A) /\
                                       (forall (C D:particular), (D ε A
/\ C ε A) -> D ε C) /\
                                       forall (K:particular), (K ε A ->
K ε a)) -> A ε a.
Proof.
intros A a H1.
destruct H1 as [H1 H2].
destruct H2 as [H2 H3].
specialize (H3 A).
apply H3.
apply A01.
Qed.

Theorem Equiv_Lesniewski_axiom : forall (A:particular)(a:plural), A ε a
<-> (exists B:particular, B ε A) /\
                                       (forall (C D:particular), (D ε A
/\ C ε A) -> D ε C) /\
                                       forall (K:particular), K ε A ->
K ε a.
Proof.
intros A a.
split.
- apply (Equiv_Lesniewski_axiom_right A a).
- eapply (Equiv_Lesniewski_axiom_left A a).
Qed.


(* ======================== particular equality
===============================*)

Lemma Tpart1 : forall A B:particular, B ε A -> A ε B. (* symmetry *)
Proof.
intros A B H1.
assert (H2:A ε A).
- apply A01.
- assert (H4:=H2);apply (A02 A A B);split;assumption.
Qed.

Lemma Tpart2 : forall A B C:particular, A ε B -> B ε C -> A ε C. (*
transitivity *)
Proof.
intros A B C H1 H3.
apply (Tpart1 C B) in H3.
apply (A02 A B C).
split;assumption.
Qed.

Axiom eqpart : forall (A B:particular), A ε B /\ B ε A <-> Eq_name A B.


Lemma eqpart_dec : forall pa1 pa2:particular, pa1 = pa2 \/ ~(pa1 = pa2).
Proof.
intros x y.
apply classic.
Qed.

Definition weakInclusion (a b :plural) := forall A:particular, A ε a ->
A ε b.
Notation "a '⊆' b" := (weakInclusion a b)  (at level 40).

Lemma LO1 : forall (A:particular)(a:plural),  A ε a -> exists
B:particular, (B ε A /\ B ε a).
Proof.
intros A a H.
exists A.
split.
- apply A01.
- assumption.
Qed.

(* characteristic thesis of LO *)
Lemma LO6 : forall (A B:particular)(a:plural),  B ε A /\ A ε a -> A ε B.
Proof.
intros A B a H1.
destruct H1 as [H1 H2].
apply Tpart1 in H1.
assumption.
Qed.

Lemma LO7 : forall (A B:particular)(a:plural),  A ε B /\ B ε a -> A ε a.
Proof.
intros A B a H1.
destruct H1 as [H1 H2].
rewrite (A03 B a) in H2.
specialize (H2 A).
apply H2 in H1.
assumption.
Qed.

Lemma LO9 : forall (A B C E:particular), ((forall D, D ε C -> B ε D) /\
(A ε C /\ E ε C)) -> A ε E.
Proof.
intros A B C E.
intros H1.
destruct H1 as [H1 H2].
destruct H2 as [H2 H3].
apply (Tpart2 A C E).
- assumption.
- apply Tpart1 in H3.
  assumption.
Qed.

Lemma LO10 : forall (A C :particular), (A ε C /\ (forall D, D ε C -> D ε
A)) -> C ε A.
Proof.
intros A C H1.
destruct H1 as [H1 H2].
apply Tpart1 in H1.
assumption.
Qed.

Lemma LO14 : forall (A:particular)(a:plural),  A ε a -> A ε A.
Proof.
intros A a H.
apply A01.
Qed.

Lemma LO15 : forall (A:particular), (exists (a:plural),  A ε a) <-> A ε A.
Proof.
intro A.
split.
+ intro H.
  apply A01.
+ intro H.
  apply A04 in H.
  assumption.
Qed.

Lemma LO16 : forall (A:particular)(a:plural), A ε a -> exists
B:particular, (A ε B /\ B ε a).
Proof.
intros A a H1.
exists A.
split.
+ apply A01.
+ trivial.
Qed.

Lemma LO24 : forall (A:particular)(b:plural),  A ε b -> forall a, (A ε a
\/ ~(A ε a)).
Proof.
intros A b H a.
apply classic.
Qed.

Lemma LO28 : forall (A:particular)(a:plural),  A ε a <-> ~~(A ε a).
Proof.
intros A a.
split.
+ intro H.
  red.
  intro H'.
  contradiction.
+ intro H.
  apply NNPP in H.
assumption.
Qed.



(* =========================== plural equality
==================================== *)


Definition Ext_Eq (a b:plural) := forall A, A ε a <-> A ε b.


Theorem Ext_Eq_rect : forall (A :particular)(a:plural), A ε a -> forall
B:particular, A ε B -> B ε a.
Proof.
intros A a H1.
intros B H2.
rewrite (A03 A a) in H1.
specialize (H1 B).
apply (Tpart1 B A) in H2.
apply H1 in H2;assumption.
Qed.

Lemma Ext_Eq_dec : forall (x y : plural), Ext_Eq x y \/ ~(Ext_Eq x y).
Proof.
intros x y.
apply classic.
Qed.


(* ================================ MEREOLOGY
==================================*)

(* element et part sous-type des pluraux *)

Class part:Type@{Name} := mkPart { pt01 : particular }.
Variable  c03 : part -> plural. Coercion c03 : part >-> plural.

Axiom TransitivePart :  forall (s1:particular)(p2 p3:part), s1 ε p2 ->
(@pt01 p2) ε p3 -> s1 ε p3.
Axiom AsymetricPart  : forall (pA pB :part),  (@pt01 pA) ε pB ->
~((@pt01 pB) ε pA).

Class el :Type@{Name} := mkEl { el01  : particular }.
Variable  c04 : el -> plural. Coercion c04 : el >-> plural.

Axiom isElement  : forall (elB:el)(A :particular), A ε elB <-> Eq_name A
(@el01 elB) \/ forall ptB:part, Eq_name (@el01 elB)(@pt01 ptB) /\ A ε ptB.

Class klass : Type@{Name}  := mkKlass {
    k01  : name;
    k02  : exists B:particular, B ε k01 }.

(* Klass sous-type des particulars *)
Variable c07 : klass -> particular. Coercion c07 : klass >-> particular.

Axiom isKlass : forall (elA:el)(ka:klass), (@el01 elA) ε ka <-> (forall
B, B ε (@k01 ka) -> B ε elA) /\
                                                  (forall (elB:el),
(@el01 elB) ε elA -> exists (elC elD:el), ((@el01 elC) ε (@k01 ka) /\
(@el01 elD) ε elC /\ (@el01 elD) ε elB)).

(* if plurals are identical then there is only one class *)

Lemma uniq_class : forall (cA cB:klass), Eq_name (@k01 cA) (@k01 cB) ->
Eq_name cA cB.
Proof.
destruct cA as [a p1], cB as [b p2].
simpl.
intro H.
revert p1 p2.
rewrite <-H.
intros p1 p2.
now rewrite (proof_irrelevance _ p1 p2).
Qed.


Lemma Th5 : forall (A:particular)(ptA:part), Eq_name A (@pt01 ptA) ->
~(A ε ptA).
Proof.
intros A ptA H.
red.
intro H'.
*rewrite H in H'. --> fail*
(*
assert (H1:=H').
apply (AsymetricPart ptA ptA) in H1.
contradiction.
*)

End Mereo.

Post by Fabian Kunze
The minimal example does not compile for me (I assume you leave
something out at "..."? ). I would have had a look at it if it would
compile.