mirror of https://github.com/nmvdw/HITs-Examples
Integers form initial ring.
This commit is contained in:
Niels
parent
6a3965dfa7
commit
8cf115c9ab
|
|
@ -1,5 +1,5 @@
|
|||
Require Import HoTT.
|
||||
Require Export HoTT.
|
||||
Require Import HoTT HitTactics.
|
||||
From HoTTClasses Require Import interfaces.abstract_algebra tactics.ring_tac theory.rings.
|
||||
|
||||
Module Export Ints.
|
||||
|
||||
|
|
@ -26,11 +26,11 @@ Module Export Ints.
|
|||
{struct x}
|
||||
: P x
|
||||
:=
|
||||
(match x return _ -> _ -> P x with
|
||||
| zero_Z => fun _ => fun _ => a
|
||||
| succ n => fun _ => fun _ => s n (Z_ind n)
|
||||
| pred n => fun _ => fun _ => p n (Z_ind n)
|
||||
end) i1 i2.
|
||||
(match x return _ -> _ -> _ -> P x with
|
||||
| zero_Z => fun _ _ _ => a
|
||||
| succ n => fun _ _ _ => s n (Z_ind n)
|
||||
| pred n => fun _ _ _ => p n (Z_ind n)
|
||||
end) i1 i2 H.
|
||||
|
||||
Axiom Z_ind_beta_inv1 : forall (n : Z), apD Z_ind (inv1 n) = i1 n (Z_ind n).
|
||||
|
||||
|
|
@ -48,7 +48,7 @@ Module Export Ints.
|
|||
|
||||
Definition Z_rec : Z -> P.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; simpl.
|
||||
simple refine (Z_ind _ _ _ _ _ _ _) ; simpl.
|
||||
- apply a.
|
||||
- intro ; apply s.
|
||||
- intro ; apply p.
|
||||
|
|
@ -75,7 +75,11 @@ Module Export Ints.
|
|||
Defined.
|
||||
|
||||
End Z_recursion.
|
||||
|
||||
|
||||
Instance FSet_recursion : HitRecursion Z :=
|
||||
{
|
||||
indTy := _; recTy := _;
|
||||
H_inductor := Z_ind; H_recursor := Z_rec }.
|
||||
End Ints.
|
||||
|
||||
Section ring_Z.
|
||||
|
|
@ -85,11 +89,11 @@ Section ring_Z.
|
|||
| S m => succ (nat_to_Z m)
|
||||
end.
|
||||
|
||||
Definition plus : Z -> Z -> Z := fun x => Z_rec Z x succ pred inv1 inv2.
|
||||
Definition plus : Z -> Z -> Z := fun x => Z_rec Z _ x succ pred inv1 inv2.
|
||||
|
||||
Lemma plus_0n : forall x, plus zero_Z x = x.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros y Hy.
|
||||
apply (ap succ Hy).
|
||||
|
|
@ -101,7 +105,7 @@ Section ring_Z.
|
|||
|
||||
Lemma plus_Sn x : forall y, plus (succ x) y = succ(plus x y).
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros y Hy.
|
||||
apply (ap succ Hy).
|
||||
|
|
@ -113,7 +117,7 @@ Section ring_Z.
|
|||
|
||||
Lemma plus_Pn x : forall y, plus (pred x) y = pred (plus x y).
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros y Hy.
|
||||
apply (ap succ Hy @ (inv2 (plus x y))^ @ inv1 (plus x y)).
|
||||
|
|
@ -125,8 +129,7 @@ Section ring_Z.
|
|||
|
||||
Lemma plus_comm x : forall y : Z, plus x y = plus y x.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _)
|
||||
; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- apply (plus_0n x)^.
|
||||
- intros n H1.
|
||||
apply (ap succ H1 @ (plus_Sn _ _)^).
|
||||
|
|
@ -136,8 +139,7 @@ Section ring_Z.
|
|||
|
||||
Lemma plus_assoc x y : forall z : Z, plus (plus x y) z = plus x (plus y z).
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _)
|
||||
; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sz HSz.
|
||||
refine (ap succ HSz).
|
||||
|
|
@ -145,11 +147,11 @@ Section ring_Z.
|
|||
apply (ap pred HPz).
|
||||
Defined.
|
||||
|
||||
Definition negate : Z -> Z := Z_rec Z zero_Z pred succ inv2 inv1.
|
||||
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).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sy HSy.
|
||||
apply (ap succ HSy).
|
||||
|
|
@ -161,7 +163,7 @@ Section ring_Z.
|
|||
|
||||
Lemma plus_negatex : forall x, plus x (negate x) = zero_Z.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sx HSx.
|
||||
refine (ap pred (plus_Sn _ _) @ _).
|
||||
|
|
@ -176,7 +178,7 @@ Section ring_Z.
|
|||
|
||||
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).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sy HSy.
|
||||
apply (ap pred HSy).
|
||||
|
|
@ -186,7 +188,7 @@ Section ring_Z.
|
|||
|
||||
Definition times (x : Z) : Z -> Z.
|
||||
Proof.
|
||||
simple refine (Z_rec _ _ _ _ _ _).
|
||||
hrecursion.
|
||||
- apply zero_Z.
|
||||
- apply (plus x).
|
||||
- apply (fun z => minus z x).
|
||||
|
|
@ -206,7 +208,7 @@ Section ring_Z.
|
|||
|
||||
Lemma times_0n : forall x, times zero_Z x = zero_Z.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
|
||||
hinduction ; try (intros ; apply set_path2) ; simpl.
|
||||
- reflexivity.
|
||||
- intros Sx HSx.
|
||||
apply (plus_0n _ @ HSx).
|
||||
|
|
@ -218,7 +220,7 @@ Section ring_Z.
|
|||
|
||||
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.
|
||||
hinduction ; try (intros ; apply set_path2) ; simpl.
|
||||
- reflexivity.
|
||||
- intros Sy HSy.
|
||||
refine (ap (fun z => plus (succ x) z) HSy @ _).
|
||||
|
|
@ -237,9 +239,15 @@ Section ring_Z.
|
|||
|
||||
Definition times_nS x y : times x (succ y) = plus x (times x y) := idpath.
|
||||
|
||||
Lemma times_1n x : times (succ zero_Z) x = x.
|
||||
Proof.
|
||||
refine (times_Sn _ _ @ _).
|
||||
refine (ap (plus x) (times_0n _) @ (plus_n0 x)).
|
||||
Defined.
|
||||
|
||||
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.
|
||||
hinduction ; try (intros ; apply set_path2) ; simpl.
|
||||
- reflexivity.
|
||||
- intros Sy HSy.
|
||||
refine (ap (fun z => plus (pred x) z) HSy @ _) ; unfold minus.
|
||||
|
|
@ -258,8 +266,7 @@ Section ring_Z.
|
|||
|
||||
Lemma times_comm x : forall y, times x y = times y x.
|
||||
Proof.
|
||||
simple refine (Z_ind _ _ _ _ _ _)
|
||||
; simpl ; try (intros ; apply set_path2).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- apply (times_0n x)^.
|
||||
- intros Sx HSx.
|
||||
apply (ap (fun z => plus x z) HSx @ (times_Sn _ _)^).
|
||||
|
|
@ -269,8 +276,7 @@ Section ring_Z.
|
|||
|
||||
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).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sy HSy.
|
||||
unfold minus.
|
||||
|
|
@ -290,8 +296,7 @@ Section ring_Z.
|
|||
|
||||
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).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sz HSz.
|
||||
refine (ap (plus x) HSz @ _).
|
||||
|
|
@ -314,8 +319,7 @@ Section ring_Z.
|
|||
|
||||
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).
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros Sz HSz.
|
||||
refine (ap (plus (times x y)) HSz @ _).
|
||||
|
|
@ -328,62 +332,153 @@ Section ring_Z.
|
|||
apply (times_negatex _ _)^.
|
||||
Defined.
|
||||
|
||||
Global Instance: Plus Z := plus.
|
||||
Global Instance: Mult Z := times.
|
||||
Global Instance: Zero Z := zero_Z.
|
||||
Global Instance: One Z := succ zero.
|
||||
Global Instance: Negate Z := negate.
|
||||
Global Instance ring_Z : Ring Z.
|
||||
Proof.
|
||||
repeat split ; try (apply _).
|
||||
- intros x y z. symmetry. apply plus_assoc.
|
||||
- intro x. apply plus_0n.
|
||||
- intro x. apply plus_xnegate.
|
||||
- intro x. apply plus_negatex.
|
||||
- intros x y. apply plus_comm.
|
||||
- intros x y z. symmetry. apply times_assoc.
|
||||
- intros x. apply times_1n.
|
||||
- intros x y. apply times_comm.
|
||||
- intros x y z.
|
||||
apply dist_times_plus.
|
||||
Defined.
|
||||
|
||||
End ring_Z.
|
||||
|
||||
Section initial_Z.
|
||||
Variable A : Type.
|
||||
Context `{Ring A}.
|
||||
|
||||
Definition ZtoA : Z -> A.
|
||||
Proof.
|
||||
hinduction ; simpl.
|
||||
- apply zero.
|
||||
- apply (Aplus one).
|
||||
- apply (Aplus (Anegate one)).
|
||||
- intros.
|
||||
symmetry.
|
||||
refine (associativity _ _ _ @ _).
|
||||
refine (ap (fun z => z & m) (left_inverse _) @ _).
|
||||
ring_with_nat.
|
||||
- intros.
|
||||
symmetry.
|
||||
refine (associativity _ _ _ @ _).
|
||||
refine (ap (fun z => z & m) (right_inverse _) @ _).
|
||||
ring_with_nat.
|
||||
Defined.
|
||||
|
||||
Lemma ZtoAplus x : forall y, ZtoA (plus x y) = Aplus (ZtoA x) (ZtoA y).
|
||||
Proof.
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- ring_with_nat.
|
||||
- intros n X.
|
||||
refine (ap (Aplus Aone) X @ _).
|
||||
ring_with_nat.
|
||||
- intros.
|
||||
refine (ap (Aplus (Anegate Aone)) X @ _).
|
||||
ring_with_nat.
|
||||
Defined.
|
||||
|
||||
Lemma ZtoAnegate : forall x, ZtoA (negate x) = Anegate (ZtoA x).
|
||||
Proof.
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- symmetry.
|
||||
apply negate_0.
|
||||
- intros n X.
|
||||
refine (ap (Aplus (Anegate Aone)) X @ _).
|
||||
symmetry.
|
||||
apply negate_plus_distr.
|
||||
- intros n X.
|
||||
refine (ap (Aplus Aone) X @ _).
|
||||
refine (ap (fun z => Aplus z (Anegate (ZtoA n))) (negate_involutive Aone)^ @ _).
|
||||
symmetry.
|
||||
apply negate_plus_distr.
|
||||
Defined.
|
||||
|
||||
Instance: SemiRingPreserving ZtoA.
|
||||
Proof.
|
||||
repeat split.
|
||||
- intro x.
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
* ring_with_nat.
|
||||
* intros y X.
|
||||
refine (ap (Aplus Aone) X @ _).
|
||||
ring_with_nat.
|
||||
* intros y X.
|
||||
refine (ap (Aplus (Anegate Aone)) X @ _).
|
||||
ring_with_nat.
|
||||
- intro x.
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
* ring_with_nat.
|
||||
* intros y X ; cbn.
|
||||
refine (ZtoAplus x _ @ _).
|
||||
refine (ap (Aplus (ZtoA x)) X @ _).
|
||||
ring_with_nat.
|
||||
* intros y X.
|
||||
cbn.
|
||||
refine (ZtoAplus _ _ @ _).
|
||||
refine (ap (Aplus _) (ZtoAnegate _) @ _).
|
||||
refine (ap (fun z => Aplus z _) X @ _).
|
||||
refine (_ @ ap (fun z => ZtoA x & z) (commutativity (ZtoA y) (Anegate Aone))).
|
||||
refine (_ @ (distribute_l (ZtoA x) _ _)^).
|
||||
refine (ap (Aplus (ZtoA x & ZtoA y)) _).
|
||||
refine (_ @ commutativity _ (ZtoA x)).
|
||||
apply negate_mult.
|
||||
- unfold UnitPreserving ; compute.
|
||||
apply H.
|
||||
Defined.
|
||||
|
||||
Theorem uniqueness (f : Z -> A) {H0 : SemiRingPreserving f} : forall x, ZtoA x = f x.
|
||||
Proof.
|
||||
assert (f (succ zero_Z) = Aone) as fone.
|
||||
{ apply H0. }
|
||||
assert (forall x y, f(plus x y) = Aplus (f x) (f y)) as fplus.
|
||||
{ apply H0. }
|
||||
compute-[times plus ZtoA] in *.
|
||||
hinduction ; simpl ; try (intros ; apply set_path2).
|
||||
- symmetry. apply H0.
|
||||
- intros x Hx.
|
||||
refine (ap (Aplus _) Hx @ _).
|
||||
refine (ap (fun z => Aplus z (f x)) fone^ @ _).
|
||||
refine ((fplus _ _)^ @ _).
|
||||
refine (ap f _).
|
||||
refine (plus_Sn _ _ @ _).
|
||||
refine (ap succ (plus_0n _)).
|
||||
- intros x Hx.
|
||||
refine (ap (Aplus _) Hx @ _).
|
||||
refine (ap (fun z => Aplus (Anegate z) (f x)) fone^ @ _).
|
||||
refine (ap (fun z => Aplus z (f x)) _ @ _).
|
||||
* symmetry. apply preserves_negate.
|
||||
* refine ((fplus _ _)^ @ _).
|
||||
refine (ap f _) ; cbn.
|
||||
refine (plus_Pn _ _ @ _).
|
||||
apply (ap pred (plus_0n x)).
|
||||
Defined.
|
||||
|
||||
End initial_Z.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
(*
|
||||
Definition Z_to_S : Z -> S1.
|
||||
Proof.
|
||||
refine (Z_rec _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 1.
|
||||
apply base.
|
||||
|
||||
Focus 3.
|
||||
intro x.
|
||||
apply x.
|
||||
|
||||
Focus 3.
|
||||
intro x.
|
||||
apply x.
|
||||
|
||||
Focus 1.
|
||||
intro m.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
refine (S1_ind _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 2.
|
||||
apply loop.
|
||||
|
||||
rewrite @HoTT.Types.Paths.transport_paths_FlFr.
|
||||
rewrite ap_idmap.
|
||||
rewrite concat_Vp.
|
||||
rewrite concat_1p.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma eq1 : ap Z_to_S (inv1 (pred (succ (pred nul)))) = reflexivity base.
|
||||
Proof.
|
||||
rewrite Z_rec_beta_inv1.
|
||||
reflexivity.
|
||||
Qed.
|
||||
|
||||
|
||||
Lemma eq2 : ap Z_to_S (ap pred (inv2 (succ (pred nul)))) = loop.
|
||||
Proof.
|
||||
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).
|
||||
Focus 2.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
rewrite Z_rec_beta_inv2.
|
||||
compute.
|
||||
reflexivity.
|
||||
Qed.
|
||||
|
||||
Module Export AltInts.
|
||||
|
||||
|
|
@ -0,0 +1,131 @@
|
|||
Require Import HoTT.
|
||||
|
||||
Module Export BadInts.
|
||||
|
||||
Private Inductive Z : Type :=
|
||||
| zero_Z : Z
|
||||
| succ : Z -> Z
|
||||
| pred : Z -> Z.
|
||||
|
||||
Axiom inv1 : forall n : Z, n = pred(succ n).
|
||||
Axiom inv2 : forall n : Z, n = succ(pred n).
|
||||
|
||||
Section Z_induction.
|
||||
Variable (P : Z -> Type)
|
||||
(a : P zero_Z)
|
||||
(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)).
|
||||
|
||||
Fixpoint Z_ind
|
||||
(x : Z)
|
||||
{struct x}
|
||||
: P x
|
||||
:=
|
||||
(match x return _ -> _ -> P x with
|
||||
| zero_Z => fun _ => fun _ => a
|
||||
| succ n => fun _ => fun _ => s n (Z_ind n)
|
||||
| pred n => fun _ => fun _ => p n (Z_ind n)
|
||||
end) i1 i2.
|
||||
|
||||
Axiom Z_ind_beta_inv1 : forall (n : Z), apD Z_ind (inv1 n) = i1 n (Z_ind n).
|
||||
|
||||
Axiom Z_ind_beta_inv2 : forall (n : Z), apD Z_ind (inv2 n) = i2 n (Z_ind n).
|
||||
End Z_induction.
|
||||
|
||||
Section Z_recursion.
|
||||
Context {P : Type}.
|
||||
Variable (a : P)
|
||||
(s : P -> P)
|
||||
(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 BadInts.
|
||||
|
||||
Section NotSet.
|
||||
Context `{Univalence}.
|
||||
|
||||
Definition Z_to_S : Z -> S1.
|
||||
Proof.
|
||||
refine (Z_rec base idmap idmap (fun _ => idpath) _).
|
||||
simple refine (S1_ind _ _ _).
|
||||
- apply loop.
|
||||
- refine (transport_paths_FlFr _ _ @ _).
|
||||
refine (ap (fun z => _ @ z) (ap_idmap _) @ _).
|
||||
refine (ap (fun z => (z^ @ loop) @ loop) (ap_idmap _) @ _).
|
||||
apply (ap (fun z => z @ loop) (concat_Vp loop) @ concat_1p loop).
|
||||
Defined.
|
||||
|
||||
Definition p1 : pred (succ (pred zero_Z)) = pred (succ (pred (succ (pred zero_Z))))
|
||||
:= inv1 (pred (succ (pred zero_Z))).
|
||||
|
||||
Lemma q1 : ap Z_to_S p1 = reflexivity base.
|
||||
Proof.
|
||||
apply Z_rec_beta_inv1.
|
||||
Defined.
|
||||
|
||||
Definition p2 : pred (succ (pred zero_Z)) = pred (succ (pred (succ (pred zero_Z))))
|
||||
:= ap pred (inv2 (succ (pred zero_Z))).
|
||||
|
||||
Lemma q2 : ap Z_to_S p2 = loop.
|
||||
Proof.
|
||||
refine ((ap_compose _ _ _)^ @ _).
|
||||
assert (forall (n m : Z) (p : n = m), ap (fun n : Z => Z_to_S(pred n)) p = ap Z_to_S p) as X.
|
||||
{ reflexivity. }
|
||||
refine (X _ _ _ @ _).
|
||||
unfold Z_to_S.
|
||||
refine (Z_rec_beta_inv2 _ _ _ _ _ (succ (pred zero_Z))).
|
||||
Defined.
|
||||
|
||||
Lemma ZSet_loop_refl (ZSet : IsHSet Z) : idpath = loop.
|
||||
Proof.
|
||||
assert (ap Z_to_S p1 = ap Z_to_S p2).
|
||||
{
|
||||
assert (p1 = p2). { apply (ZSet _ _ p1 p2). }
|
||||
apply (ap (fun z => ap Z_to_S z) X).
|
||||
}
|
||||
apply (q1^ @ X @ q2).
|
||||
Defined.
|
||||
|
||||
Lemma ZSet_not_hset (ZSet : IsHSet Z) : False.
|
||||
Proof.
|
||||
enough (idpath = loop).
|
||||
- assert (S1_encode _ idpath = S1_encode _ (loopexp loop (pos Int.one))) as H' by f_ap.
|
||||
rewrite S1_encode_loopexp in H'. simpl in H'. symmetry in H'.
|
||||
apply (pos_neq_zero H').
|
||||
- apply ZSet_loop_refl.
|
||||
apply ZSet.
|
||||
Qed.
|
||||
End NotSet.
|
||||
Loading…
Reference in New Issue