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Integers form a ring

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Niels 2017-08-04 14:46:08 +02:00
parent 6f016d1b7f
commit 376efbf2e9
1 changed files with 510 additions and 307 deletions
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Integers.v
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@ -3,135 +3,337 @@ Require Export HoTT.
Module Export Ints. Module Export Ints.
Private Inductive Z : Type0 := Private Inductive Z : Type :=
| nul : Z | zero_Z : Z
| succ : Z -> Z | succ : Z -> Z
| pred : Z -> Z. | pred : Z -> Z.
Axiom inv1 : forall n : Z, n = pred(succ n). Axiom inv1 : forall n : Z, n = pred(succ n).
Axiom inv2 : forall n : Z, n = succ(pred n). Axiom inv2 : forall n : Z, n = succ(pred n).
Axiom ZisSet : IsHSet Z.
Fixpoint Z_rec Section Z_induction.
(P : Type) Variable (P : Z -> Type)
(a : P) (H : forall n, IsHSet (P n))
(s : P -> P) (a : P zero_Z)
(p : P -> P)
(i1 : forall (m : P), m = p(s m))
(i2 : forall (m : P), m = s(p m))
(x : Z)
{struct x}
: P
:=
(match x return _ -> _ -> P with
| nul => fun _ => fun _ => a
| succ n => fun _ => fun _ => s ((Z_rec P a s p i1 i2) n)
| pred n => fun _ => fun _ => p ((Z_rec P a s p i1 i2) n)
end) i1 i2.
Axiom Z_rec_beta_inv1 : forall
(P : Type)
(a : P)
(s : P -> P)
(p : P -> P)
(i1 : forall (m : P), m = p(s m))
(i2 : forall (m : P), m = s(p m))
(n : Z)
, ap (Z_rec P a s p i1 i2) (inv1 n) = i1 (Z_rec P a s p i1 i2 n).
Axiom Z_rec_beta_inv2 : forall
(P : Type)
(a : P)
(s : P -> P)
(p : P -> P)
(i1 : forall (m : P), m = p(s m))
(i2 : forall (m : P), m = s(p m))
(n : Z)
, ap (Z_rec P a s p i1 i2) (inv2 n) = i2 (Z_rec P a s p i1 i2 n).
Fixpoint Z_ind
(P : Z -> Type)
(a : P nul)
(s : forall (n : Z), P n -> P (succ n)) (s : forall (n : Z), P n -> P (succ n))
(p : forall (n : Z), P n -> P (pred n)) (p : forall (n : Z), P n -> P (pred n))
(i1 : forall (n : Z) (m : P n), (inv1 n) # m = p (succ n) (s (n) m)) (i1 : forall (n : Z) (m : P n), (inv1 n) # m = p (succ n) (s (n) m))
(i2 : forall (n : Z) (m : P n), (inv2 n) # m = s (pred n) (p (n) m)) (i2 : forall (n : Z) (m : P n), (inv2 n) # m = s (pred n) (p (n) m)).
Fixpoint Z_ind
(x : Z) (x : Z)
{struct x} {struct x}
: P x : P x
:= :=
(match x return _ -> _ -> P x with (match x return _ -> _ -> P x with
| nul => fun _ => fun _ => a | zero_Z => fun _ => fun _ => a
| succ n => fun _ => fun _ => s n ((Z_ind P a s p i1 i2) n) | succ n => fun _ => fun _ => s n (Z_ind n)
| pred n => fun _ => fun _ => p n ((Z_ind P a s p i1 i2) n) | pred n => fun _ => fun _ => p n (Z_ind n)
end) i1 i2. end) i1 i2.
Axiom Z_ind_beta_inv1 : forall Axiom Z_ind_beta_inv1 : forall (n : Z), apD Z_ind (inv1 n) = i1 n (Z_ind n).
(P : Z -> Type)
(a : P nul)
(s : forall (n : Z), P n -> P (succ n))
(p : forall (n : Z), P n -> P (pred n))
(i1 : forall (n : Z) (m : P n), (inv1 n) # m = p (succ n) (s (n) m))
(i2 : forall (n : Z) (m : P n), (inv2 n) # m = s (pred n) (p (n) m))
(n : Z)
, apD (Z_ind P a s p i1 i2) (inv1 n) = i1 n (Z_ind P a s p i1 i2 n).
Axiom Z_ind_beta_inv2 : forall Axiom Z_ind_beta_inv2 : forall (n : Z), apD Z_ind (inv2 n) = i2 n (Z_ind n).
(P : Z -> Type) End Z_induction.
(a : P nul)
(s : forall (n : Z), P n -> P (succ n)) Section Z_recursion.
(p : forall (n : Z), P n -> P (pred n)) Variable (P : Type)
(i1 : forall (n : Z) (m : P n), (inv1 n) # m = p (succ n) (s (n) m)) (H : IsHSet P)
(i2 : forall (n : Z) (m : P n), (inv2 n) # m = s (pred n) (p (n) m)) (a : P)
(n : Z) (s : P -> P)
, apD (Z_ind P a s p i1 i2) (inv2 n) = i2 n (Z_ind P a s p i1 i2 n). (p : P -> P)
(i1 : forall (m : P), m = p(s m))
(i2 : forall (m : P), m = s(p m)).
Definition Z_rec : Z -> P.
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl.
- apply a.
- intro ; apply s.
- intro ; apply p.
- intros.
refine (transport_const _ _ @ (i1 _)).
- intros.
refine (transport_const _ _ @ (i2 _)).
Defined.
Definition Z_rec_beta_inv1 (n : Z) : ap Z_rec (inv1 n) = i1 (Z_rec n).
Proof.
unfold Z_rec.
eapply (cancelL (transport_const (inv1 n) _)).
simple refine ((apD_const _ _)^ @ _).
apply Z_ind_beta_inv1.
Defined.
Definition Z_rec_beta_inv2 (n : Z) : ap Z_rec (inv2 n) = i2 (Z_rec n).
Proof.
unfold Z_rec.
eapply (cancelL (transport_const (inv2 n) _)).
simple refine ((apD_const _ _)^ @ _).
apply Z_ind_beta_inv2.
Defined.
End Z_recursion.
End Ints. End Ints.
Definition negate : Z -> Z. Section ring_Z.
Proof. Fixpoint nat_to_Z (n : nat) : Z :=
intro x. match n with
refine (Z_rec _ _ _ _ _ _ x). | 0 => zero_Z
Unshelve. | S m => succ (nat_to_Z m)
Focus 1. end.
apply nul.
Focus 3. Definition plus : Z -> Z -> Z := fun x => Z_rec Z x succ pred inv1 inv2.
apply pred.
Focus 3. Lemma plus_0n : forall x, plus zero_Z x = x.
apply succ. Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros y Hy.
apply (ap succ Hy).
- intros y Hy.
apply (ap pred Hy).
Defined.
Focus 2. Definition plus_n0 x : plus x zero_Z = x := idpath x.
apply inv1.
apply inv2. Lemma plus_Sn x : forall y, plus (succ x) y = succ(plus x y).
Defined. Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros y Hy.
apply (ap succ Hy).
- intros y Hy.
apply (ap pred Hy @ (inv1 (plus x y))^ @ inv2 (plus x y)).
Defined.
Definition plus : Z -> Z -> Z. Definition plus_nS x y : plus x (succ y) = succ(plus x y) := idpath.
Proof.
intro x.
refine (Z_rec _ _ _ _ _ _).
Unshelve.
Focus 1.
apply x.
Focus 3. Lemma plus_Pn x : forall y, plus (pred x) y = pred (plus x y).
apply succ. Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros y Hy.
apply (ap succ Hy @ (inv2 (plus x y))^ @ inv1 (plus x y)).
- intros y Hy.
apply (ap pred Hy).
Defined.
Focus 3. Definition plus_nP x y : plus x (pred y) = pred(plus x y) := idpath.
apply pred.
Focus 1. Lemma plus_comm x : forall y : Z, plus x y = plus y x.
apply inv1. Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- apply (plus_0n x)^.
- intros n H1.
apply (ap succ H1 @ (plus_Sn _ _)^).
- intros n H1.
apply (ap pred H1 @ (plus_Pn _ _)^).
Defined.
apply inv2. Lemma plus_assoc x y : forall z : Z, plus (plus x y) z = plus x (plus y z).
Defined. Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sz HSz.
refine (ap succ HSz).
- intros Pz HPz.
apply (ap pred HPz).
Defined.
Definition minus (x : Z) (y : Z) := plus x (negate y). Definition negate : Z -> Z := Z_rec Z zero_Z pred succ inv2 inv1.
Lemma negate_negate : forall x, negate(negate x) = x.
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sy HSy.
apply (ap succ HSy).
- intros Py HPy.
apply (ap pred HPy).
Defined.
Definition minus x y : Z := plus x (negate y).
Lemma plus_negatex : forall x, plus x (negate x) = zero_Z.
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sx HSx.
refine (ap pred (plus_Sn _ _) @ _).
refine ((inv1 _)^ @ HSx).
- intros Px HPx.
refine (ap succ (plus_Pn _ _) @ _).
refine ((inv2 _)^ @ HPx).
Defined.
Definition plus_xnegate x : plus (negate x) x = zero_Z :=
plus_comm (negate x) x @ plus_negatex x.
Lemma plus_negate x : forall y, plus (negate x) (negate y) = negate (plus x y).
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sy HSy.
apply (ap pred HSy).
- intros Py HPy.
apply (ap succ HPy).
Defined.
Definition times (x : Z) : Z -> Z.
Proof.
simple refine (Z_rec _ _ _ _ _ _).
- apply zero_Z.
- apply (plus x).
- apply (fun z => minus z x).
- intros ; unfold minus.
symmetry.
refine (ap (fun z => plus z (negate x)) (plus_comm x m) @ _).
refine (plus_assoc _ _ _ @ _).
refine (ap (fun z => plus m z) (plus_negatex _) @ _).
apply plus_n0.
- intros ; unfold minus.
symmetry.
refine (ap (fun z => plus x z) (plus_comm _ _) @ _).
refine ((plus_assoc _ _ _)^ @ _).
refine (ap (fun z => plus z m) (plus_negatex _) @ _).
apply plus_0n.
Defined.
Lemma times_0n : forall x, times zero_Z x = zero_Z.
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
- reflexivity.
- intros Sx HSx.
apply (plus_0n _ @ HSx).
- intros Px HPx.
unfold minus ; simpl ; apply HPx.
Defined.
Definition times_n0 n : times n zero_Z = zero_Z := idpath.
Lemma times_Sn x : forall y, times (succ x) y = plus y (times x y).
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
- reflexivity.
- intros Sy HSy.
refine (ap (fun z => plus (succ x) z) HSy @ _).
refine (plus_Sn _ _ @ _).
refine (_ @ (plus_Sn _ _)^).
refine (ap succ _).
refine ((plus_assoc _ _ _)^ @ _).
refine (_ @ plus_assoc _ _ _).
refine (ap (fun z => plus z (times x Sy)) (plus_comm _ _)).
- intros Py HPy ; unfold minus.
refine (ap (fun z => plus z (negate (succ x))) HPy @ _) ; simpl.
refine (_ @ (plus_Pn _ _)^).
refine (ap pred _).
apply plus_assoc.
Defined.
Definition times_nS x y : times x (succ y) = plus x (times x y) := idpath.
Lemma times_Pn x : forall y, times (pred x) y = minus (times x y) y.
Proof.
simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
- reflexivity.
- intros Sy HSy.
refine (ap (fun z => plus (pred x) z) HSy @ _) ; unfold minus.
refine (plus_Pn _ _ @ _) ; simpl.
refine (ap pred _).
apply (plus_assoc _ _ _)^.
- intros Py HPy.
refine (ap (fun z => minus z (pred x)) HPy @ _) ; unfold minus ; simpl.
refine (ap succ _).
refine (plus_assoc _ _ _ @ _).
refine (_ @ (plus_assoc _ _ _)^).
refine (ap (fun z => plus (times x Py) z) (plus_comm _ _)).
Defined.
Definition times_nP x y : times x (pred y) = minus (times x y) x := idpath.
Lemma times_comm x : forall y, times x y = times y x.
Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- apply (times_0n x)^.
- intros Sx HSx.
apply (ap (fun z => plus x z) HSx @ (times_Sn _ _)^).
- intros Py HPy.
apply (ap (fun z => minus z x) HPy @ (times_Pn _ _)^).
Defined.
Lemma times_negatex x : forall y, times x (negate y) = negate (times x y).
Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sy HSy.
unfold minus.
refine (ap (fun z => plus z (negate x)) HSy @ _).
refine (plus_negate _ _ @ _).
apply (ap negate (plus_comm _ _)).
- intros Py HPy.
refine (ap (plus x) HPy @ _).
unfold minus.
refine (ap (fun z => plus z (negate (times x Py))) (negate_negate _)^ @ _).
refine (plus_negate _ _ @ _).
refine (ap negate (plus_comm _ _)).
Defined.
Definition times_xnegate x y : times (negate x) y = negate (times x y) :=
times_comm (negate x) y @ times_negatex y x @ ap negate (times_comm y x).
Lemma dist_times_plus x y : forall z, times x (plus y z) = plus (times x y) (times x z).
Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sz HSz.
refine (ap (plus x) HSz @ _).
refine ((plus_assoc _ _ _)^ @ _).
refine (_ @ plus_assoc _ _ _).
refine (ap (fun z => plus z (times x Sz)) (plus_comm _ _)).
- intros Pz HPz.
refine (ap (fun z => minus z x) HPz @ _).
unfold minus ; simpl.
apply plus_assoc.
Defined.
Lemma dist_plus_times x y z : times (plus x y) z = plus (times x z) (times y z).
Proof.
refine (times_comm _ _ @ _).
refine (dist_times_plus _ _ _ @ _).
refine (ap (plus (times z x)) (times_comm _ _) @ _).
apply (ap (fun a => plus a (times y z)) (times_comm _ _)).
Defined.
Lemma times_assoc x y : forall z, times (times x y) z = times x (times y z).
Proof.
simple refine (Z_ind _ _ _ _ _ _)
; simpl ; try (intros ; apply set_path2).
- reflexivity.
- intros Sz HSz.
refine (ap (plus (times x y)) HSz @ _).
symmetry ; apply dist_times_plus.
- intros Pz HPz.
refine (ap (fun z => minus z (times x y)) HPz @ _).
unfold minus.
refine (_ @ (dist_times_plus _ _ _)^).
refine (ap (plus (times x (times y Pz))) _).
apply (times_negatex _ _)^.
Defined.
End ring_Z.
(*
Definition Z_to_S : Z -> S1. Definition Z_to_S : Z -> S1.
Proof. Proof.
refine (Z_rec _ _ _ _ _ _). refine (Z_rec _ _ _ _ _ _).
Unshelve. Unshelve.
Focus 1. Focus 1.
@ -165,15 +367,15 @@ Defined.
Lemma eq1 : ap Z_to_S (inv1 (pred (succ (pred nul)))) = reflexivity base. Lemma eq1 : ap Z_to_S (inv1 (pred (succ (pred nul)))) = reflexivity base.
Proof. Proof.
rewrite Z_rec_beta_inv1. rewrite Z_rec_beta_inv1.
reflexivity. reflexivity.
Qed. Qed.
Lemma eq2 : ap Z_to_S (ap pred (inv2 (succ (pred nul)))) = loop. Lemma eq2 : ap Z_to_S (ap pred (inv2 (succ (pred nul)))) = loop.
Proof. Proof.
rewrite <- ap_compose. rewrite <- ap_compose.
enough (forall (n m : Z) (p : n = m), ap Z_to_S p = ap (fun n : Z => Z_to_S(pred n)) p). enough (forall (n m : Z) (p : n = m), ap Z_to_S p = ap (fun n : Z => Z_to_S(pred n)) p).
Focus 2. Focus 2.
compute. compute.
reflexivity. reflexivity.
@ -185,13 +387,13 @@ Qed.
Module Export AltInts. Module Export AltInts.
Private Inductive Z' : Type0 := Private Inductive Z' : Type0 :=
| positive : nat -> Z' | positive : nat -> Z'
| negative : nat -> Z'. | negative : nat -> Z'.
Axiom path : positive 0 = negative 0. Axiom path : positive 0 = negative 0.
Fixpoint Z'_ind Fixpoint Z'_ind
(P : Z' -> Type) (P : Z' -> Type)
(po : forall (x : nat), P (positive x)) (po : forall (x : nat), P (positive x))
(ne : forall (x : nat), P (negative x)) (ne : forall (x : nat), P (negative x))
@ -205,7 +407,7 @@ Fixpoint Z'_ind
| negative n => fun _ => ne n | negative n => fun _ => ne n
end) i. end) i.
Axiom Z'_ind_beta_inv1 : forall Axiom Z'_ind_beta_inv1 : forall
(P : Z' -> Type) (P : Z' -> Type)
(po : forall (x : nat), P (positive x)) (po : forall (x : nat), P (positive x))
(ne : forall (x : nat), P (negative x)) (ne : forall (x : nat), P (negative x))
@ -215,7 +417,7 @@ End AltInts.
Definition succ_Z' : Z' -> Z'. Definition succ_Z' : Z' -> Z'.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 2. Focus 2.
intro n. intro n.
@ -235,7 +437,7 @@ Defined.
Definition pred_Z' : Z' -> Z'. Definition pred_Z' : Z' -> Z'.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 2. Focus 2.
intro n. intro n.
@ -262,7 +464,7 @@ Fixpoint Nat_to_Pos (n : nat) : Pos :=
Definition Z'_to_Int : Z' -> Int. Definition Z'_to_Int : Z' -> Int.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 2. Focus 2.
@ -284,63 +486,63 @@ Defined.
Definition Pos_to_Nat : Pos -> nat. Definition Pos_to_Nat : Pos -> nat.
Proof. Proof.
intro x. intro x.
induction x. induction x.
apply 1. apply 1.
apply (S IHx). apply (S IHx).
Defined. Defined.
Definition Int_to_Z' (x : Int) : Z'. Definition Int_to_Z' (x : Int) : Z'.
Proof. Proof.
induction x. induction x.
apply (negative (Pos_to_Nat p)). apply (negative (Pos_to_Nat p)).
apply (positive 0). apply (positive 0).
apply (positive (Pos_to_Nat p)). apply (positive (Pos_to_Nat p)).
Defined. Defined.
Lemma Z'_to_int_pos_homomorphism : Lemma Z'_to_int_pos_homomorphism :
forall n : nat, Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)). forall n : nat, Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)).
Proof. Proof.
intro n. intro n.
reflexivity. reflexivity.
Qed. Qed.
Lemma Z'_to_int_neg_homomorphism : Lemma Z'_to_int_neg_homomorphism :
forall n : nat, Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)). forall n : nat, Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)).
Proof. Proof.
intro n. intro n.
reflexivity. reflexivity.
Qed. Qed.
Theorem isoEq1 : forall x : Int, Z'_to_Int(Int_to_Z' x) = x. Theorem isoEq1 : forall x : Int, Z'_to_Int(Int_to_Z' x) = x.
Proof. Proof.
intro x. intro x.
induction x. induction x.
induction p. induction p.
reflexivity. reflexivity.
rewrite Z'_to_int_neg_homomorphism. rewrite Z'_to_int_neg_homomorphism.
rewrite IHp. rewrite IHp.
reflexivity. reflexivity.
reflexivity. reflexivity.
induction p. induction p.
reflexivity. reflexivity.
rewrite Z'_to_int_pos_homomorphism. rewrite Z'_to_int_pos_homomorphism.
rewrite IHp. rewrite IHp.
reflexivity. reflexivity.
Defined. Defined.
Lemma Int_to_Z'_succ_homomorphism : Lemma Int_to_Z'_succ_homomorphism :
forall x : Int, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x). forall x : Int, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x).
Proof. Proof.
simpl. simpl.
intro x. intro x.
simpl. simpl.
induction x. induction x.
induction p. induction p.
compute. compute.
apply path. apply path.
@ -358,8 +560,8 @@ Qed.
Lemma Int_to_Z'_pred_homomorphism : Lemma Int_to_Z'_pred_homomorphism :
forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x). forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x).
Proof. Proof.
intro x. intro x.
induction x. induction x.
induction p. induction p.
reflexivity. reflexivity.
@ -376,7 +578,7 @@ Qed.
Theorem isoEq2 : forall x : Z', Int_to_Z'(Z'_to_Int x) = x. Theorem isoEq2 : forall x : Z', Int_to_Z'(Z'_to_Int x) = x.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 2. Focus 2.
@ -414,20 +616,20 @@ Defined.
Theorem adj : Theorem adj :
forall x : Z', isoEq1 (Z'_to_Int x) = ap Z'_to_Int (isoEq2 x). forall x : Z', isoEq1 (Z'_to_Int x) = ap Z'_to_Int (isoEq2 x).
Proof. Proof.
intro x. intro x.
apply hset_int. apply hset_int.
Defined. Defined.
Definition isomorphism : IsEquiv Z'_to_Int. Definition isomorphism : IsEquiv Z'_to_Int.
Proof. Proof.
apply (BuildIsEquiv Z' Int Z'_to_Int Int_to_Z' isoEq1 isoEq2 adj). apply (BuildIsEquiv Z' Int Z'_to_Int Int_to_Z' isoEq1 isoEq2 adj).
Qed. Qed.
Axiom everythingSet : forall T : Type, IsHSet T. Axiom everythingSet : forall T : Type, IsHSet T.
Definition Z_to_Z' : Z -> Z'. Definition Z_to_Z' : Z -> Z'.
Proof. Proof.
refine (Z_rec _ _ _ _ _ _). refine (Z_rec _ _ _ _ _ _).
Unshelve. Unshelve.
Focus 1. Focus 1.
apply (positive 0). apply (positive 0).
@ -477,7 +679,7 @@ Defined.
Definition Z'_to_Z : Z' -> Z. Definition Z'_to_Z : Z' -> Z.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 2. Focus 2.
induction 1. induction 1.
@ -500,7 +702,7 @@ Defined.
Theorem isoZEq1 : forall n : Z', Z_to_Z'(Z'_to_Z n) = n. Theorem isoZEq1 : forall n : Z', Z_to_Z'(Z'_to_Z n) = n.
Proof. Proof.
refine (Z'_ind _ _ _ _). refine (Z'_ind _ _ _ _).
Unshelve. Unshelve.
Focus 3. Focus 3.
intros. intros.
@ -542,7 +744,7 @@ Defined.
Theorem isoZEq2 : forall n : Z, Z'_to_Z(Z_to_Z' n) = n. Theorem isoZEq2 : forall n : Z, Z'_to_Z(Z_to_Z' n) = n.
Proof. Proof.
refine (Z_ind _ _ _ _ _ _). refine (Z_ind _ _ _ _ _ _).
Unshelve. Unshelve.
Focus 1. Focus 1.
reflexivity. reflexivity.
@ -622,11 +824,12 @@ Defined.
Theorem adj2 : Theorem adj2 :
forall x : Z', isoZEq2 (Z'_to_Z x) = ap Z'_to_Z (isoZEq1 x). forall x : Z', isoZEq2 (Z'_to_Z x) = ap Z'_to_Z (isoZEq1 x).
Proof. Proof.
intro x. intro x.
apply everythingSet. apply everythingSet.
Defined. Defined.
Definition isomorphism2 : IsEquiv Z'_to_Z. Definition isomorphism2 : IsEquiv Z'_to_Z.
Proof. Proof.
apply (BuildIsEquiv Z' Z Z'_to_Z Z_to_Z' isoZEq2 isoZEq1 adj2). apply (BuildIsEquiv Z' Z Z'_to_Z Z_to_Z' isoZEq2 isoZEq1 adj2).
Qed. Qed.
*)