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FinSets.v Normal file
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Require Import HoTT.
Require Export HoTT.
Module Export FinSet.
Private Inductive FinSets (A : Type) : Type :=
| empty : FinSets A
| L : A -> FinSets A
| U : FinSets A -> FinSets A -> FinSets A.
Axiom assoc : forall (A : Type) (x y z : FinSets A),
U A x (U A y z) = U A (U A x y) z.
Axiom comm : forall (A : Type) (x y : FinSets A),
U A x y = U A y x.
Axiom nl : forall (A : Type) (x : FinSets A),
U A (empty A) x = x.
Axiom nr : forall (A : Type) (x : FinSets A),
U A x (empty A) = x.
Axiom idem : forall (A : Type) (x : A),
U A (L A x) (L A x) = L A x.
Fixpoint FinSets_rec
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FinSets A)
{struct x}
: P
:= (match x return _ -> _ -> _ -> _ -> _ -> P with
| empty => fun _ => fun _ => fun _ => fun _ => fun _ => e
| L a => fun _ => fun _ => fun _ => fun _ => fun _ => l a
| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => u
(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
(FinSets_rec A P e l u assocP commP nlP nrP idemP z)
end) assocP commP nlP nrP idemP.
Axiom FinSets_beta_assoc : forall
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x y z : FinSets A),
ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (assoc A x y z)
=
(assocP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
(FinSets_rec A P e l u assocP commP nlP nrP idemP z)
).
Axiom FinSets_beta_comm : forall
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x y : FinSets A),
ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (comm A x y)
=
(commP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
).
Axiom FinSets_beta_nl : forall
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FinSets A),
ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (nl A x)
=
(nlP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
).
Axiom FinSets_beta_nr : forall
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FinSets A),
ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (nr A x)
=
(nrP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
).
Axiom FinSets_beta_idem : forall
(A : Type)
(P : Type)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : A),
ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (idem A x)
=
idemP x.
End FinSet.
Definition isIn : forall
(A : Type)
(eq : A -> A -> Bool),
A -> FinSets A -> Bool.
Proof.
intro A.
intro eq.
intro a.
refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
Unshelve.
Focus 6.
apply false.
Focus 6.
intro a'.
apply (eq a a').
Focus 6.
intro b.
intro b'.
apply (orb b b').
Focus 3.
intros.
compute.
reflexivity.
Focus 1.
intros.
compute.
destruct x.
reflexivity.
destruct y.
reflexivity.
reflexivity.
Focus 1.
intros.
compute.
destruct x.
destruct y.
reflexivity.
reflexivity.
destruct y.
reflexivity.
reflexivity.
Focus 1.
intros.
compute.
destruct x.
reflexivity.
reflexivity.
intros.
compute.
destruct (eq a x).
reflexivity.
reflexivity.
Defined.
Definition comprehension : forall
(A : Type)
(eq : A -> A -> Bool),
(A -> Bool) -> FinSets A -> FinSets A.
Proof.
intro A.
intro eq.
intro phi.
refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
Unshelve.
Focus 6.
apply empty.
Focus 6.
intro a.
apply (if (phi a) then L A a else (empty A)).
Focus 6.
intro x.
intro y.
apply (U A x y).
Focus 3.
intros.
compute.
apply nl.
Focus 1.
intros.
compute.
apply assoc.
Focus 1.
intros.
compute.
apply comm.
Focus 1.
intros.
compute.
apply nr.
intros.
compute.
destruct (phi x).
apply idem.
apply nl.
Defined.
Definition intersection : forall (A : Type) (eq : A -> A -> Bool),
FinSets A -> FinSets A -> FinSets A.
Proof.
intro A.
intro eq.
intro x.
intro y.
apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
Defined.
Definition subset (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool.
Proof.
refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
Unshelve.
Focus 6.
apply x.
Focus 6.
apply true.
Focus 6.
intro a.
apply (isIn A eq a y).
Focus 6.
intro b.
intro b'.
apply (andb b b').
Focus 1.
intros.
compute.
destruct x0.
destruct y0.
reflexivity.
reflexivity.
reflexivity.
Focus 1.
intros.
compute.
destruct x0.
destruct y0.
reflexivity.
reflexivity.
destruct y0.
reflexivity.
reflexivity.
Focus 1.
intros.
compute.
reflexivity.
Focus 1.
intros.
compute.
destruct x0.
reflexivity.
reflexivity.
intros.
destruct (isIn A eq x0 y).
compute.
reflexivity.
compute.
reflexivity.
Defined.
Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool
:= andb (subset A eq x y) (subset A eq y x).
Fixpoint eq_nat n m : Bool :=
match n, m with
| O, O => true
| O, S _ => false
| S _, O => false
| S n1, S m1 => eq_nat n1 m1
end.

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Integers.v Normal file
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Require Import HoTT.
Require Export HoTT.
Theorem useful :
forall (A B : Type)
(f g : A -> B)
(a a' : A)
(p : a = a')
(q : f a = g a),
transport (fun x => f x = g x) p q = (ap f p)^ @ q @ (ap g p).
Proof.
intros.
induction p.
rewrite transport_1.
rewrite ap_1.
rewrite ap_1.
rewrite concat_p1.
simpl.
rewrite concat_1p.
reflexivity.
Qed.
Module Export Ints.
Private Inductive Z : Type0 :=
| nul : 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).
Fixpoint Z_rec
(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))
(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))
(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))
(x : Z)
{struct x}
: P x
:=
(match x return _ -> _ -> P x with
| nul => fun _ => fun _ => a
| succ n => fun _ => fun _ => s n ((Z_ind P a s p i1 i2) n)
| pred n => fun _ => fun _ => p n ((Z_ind P a s p i1 i2) n)
end) i1 i2.
Axiom Z_ind_beta_inv1 : forall
(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
(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) (inv2 n) = i2 n (Z_ind P a s p i1 i2 n).
End Ints.
Definition negate : Z -> Z.
Proof.
intro x.
refine (Z_rec _ _ _ _ _ _ x).
Unshelve.
Focus 1.
apply nul.
Focus 3.
apply pred.
Focus 3.
apply succ.
Focus 2.
apply inv1.
apply inv2.
Defined.
Definition plus : Z -> Z -> Z.
Proof.
intro x.
refine (Z_rec _ _ _ _ _ _).
Unshelve.
Focus 1.
apply x.
Focus 3.
apply succ.
Focus 3.
apply pred.
Focus 1.
apply inv1.
apply inv2.
Defined.
Definition minus (x : Z) (y : Z) := plus x (negate y).
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 useful.
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.
Private Inductive Z' : Type0 :=
| positive : nat -> Z'
| negative : nat -> Z'.
Axiom path : positive 0 = negative 0.
Fixpoint Z'_ind
(P : Z' -> Type)
(po : forall (x : nat), P (positive x))
(ne : forall (x : nat), P (negative x))
(i : path # (po 0) = ne 0)
(x : Z')
{struct x}
: P x
:=
(match x return _ -> P x with
| positive n => fun _ => po n
| negative n => fun _ => ne n
end) i.
Axiom Z'_ind_beta_inv1 : forall
(P : Z' -> Type)
(po : forall (x : nat), P (positive x))
(ne : forall (x : nat), P (negative x))
(i : path # (po 0) = ne 0)
, apD (Z'_ind P po ne i) path = i.
End AltInts.
Definition succ_Z' : Z' -> Z'.
Proof.
refine (Z'_ind _ _ _ _).
Unshelve.
Focus 2.
intro n.
apply (positive (S n)).
Focus 2.
intro n.
induction n.
apply (positive 1).
apply (negative n).
simpl.
rewrite transport_const.
reflexivity.
Defined.
Definition pred_Z' : Z' -> Z'.
Proof.
refine (Z'_ind _ _ _ _).
Unshelve.
Focus 2.
intro n.
induction n.
apply (negative 1).
apply (positive n).
Focus 2.
intro n.
apply (negative (S n)).
simpl.
rewrite transport_const.
reflexivity.
Defined.
Fixpoint Nat_to_Pos (n : nat) : Pos :=
match n with
| 0 => Int.one
| S k => succ_pos (Nat_to_Pos k)
end.
Definition Z'_to_Int : Z' -> Int.
Proof.
refine (Z'_ind _ _ _ _).
Unshelve.
Focus 2.
intro x.
induction x.
apply (Int.zero).
apply (succ_int IHx).
Focus 2.
intro x.
induction x.
apply (Int.zero).
apply (pred_int IHx).
simpl.
rewrite transport_const.
reflexivity.
Defined.
Definition Pos_to_Nat : Pos -> nat.
Proof.
intro x.
induction x.
apply 1.
apply (S IHx).
Defined.
Definition Int_to_Z' (x : Int) : Z'.
Proof.
induction x.
apply (negative (Pos_to_Nat p)).
apply (positive 0).
apply (positive (Pos_to_Nat p)).
Defined.
Lemma Z'_to_int_pos_homomorphism :
forall n : nat, Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)).
Proof.
intro n.
reflexivity.
Qed.
Lemma Z'_to_int_neg_homomorphism :
forall n : nat, Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)).
Proof.
intro n.
reflexivity.
Qed.
Theorem isoEq1 : forall x : Int, Z'_to_Int(Int_to_Z' x) = x.
Proof.
intro x.
induction x.
induction p.
reflexivity.
rewrite Z'_to_int_neg_homomorphism.
rewrite IHp.
reflexivity.
reflexivity.
induction p.
reflexivity.
rewrite Z'_to_int_pos_homomorphism.
rewrite IHp.
reflexivity.
Defined.
Lemma Int_to_Z'_succ_homomorphism :
forall x : Int, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x).
Proof.
simpl.
intro x.
simpl.
induction x.
induction p.
compute.
apply path.
reflexivity.
reflexivity.
induction p.
reflexivity.
reflexivity.
Qed.
Lemma Int_to_Z'_pred_homomorphism :
forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x).
Proof.
intro x.
induction x.
induction p.
reflexivity.
reflexivity.
reflexivity.
induction p.
reflexivity.
reflexivity.
Qed.
Theorem isoEq2 : forall x : Z', Int_to_Z'(Z'_to_Int x) = x.
Proof.
refine (Z'_ind _ _ _ _).
Unshelve.
Focus 2.
intro x.
induction x.
reflexivity.
rewrite Z'_to_int_pos_homomorphism.
rewrite Int_to_Z'_succ_homomorphism.
rewrite IHx.
reflexivity.
Focus 2.
intro x.
induction x.
apply path.
rewrite Z'_to_int_neg_homomorphism.
rewrite Int_to_Z'_pred_homomorphism.
rewrite IHx.
reflexivity.
simpl.
rewrite useful.
rewrite concat_p1.
rewrite ap_idmap.
enough (ap (fun x : Z' => Z'_to_Int x) path = reflexivity Int.zero).
rewrite ap_compose.
rewrite X.
apply concat_1p.
apply axiomK_hset.
apply hset_int.
Defined.
Theorem adj :
forall x : Z', isoEq1 (Z'_to_Int x) = ap Z'_to_Int (isoEq2 x).
Proof.
intro x.
apply hset_int.
Defined.
Definition isomorphism : IsEquiv Z'_to_Int.
Proof.
apply (BuildIsEquiv Z' Int Z'_to_Int Int_to_Z' isoEq1 isoEq2 adj).
Qed.

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Require Import HoTT.
Require Export HoTT.
Theorem useful :
forall (A B : Type)
(f g : A -> B)
(a a' : A)
(p : a = a')
(q : f a = g a),
transport (fun x => f x = g x) p q = (ap f p)^ @ q @ (ap g p).
Proof.
intros.
induction p.
rewrite transport_1.
rewrite ap_1.
rewrite ap_1.
rewrite concat_p1.
simpl.
rewrite concat_1p.
reflexivity.
Qed.
Module Export modulo.
Private Inductive Mod2 : Type0 :=
| Z : Mod2
| succ : Mod2 -> Mod2.
Axiom mod : Z = succ(succ Z).
Fixpoint Mod2_ind
(P : Mod2 -> Type)
(a : P Z)
(s : forall (n : Mod2), P n -> P (succ n))
(mod' : mod # a = s (succ Z) (s Z a))
(x : Mod2)
{struct x}
: P x
:=
(match x return _ -> P x with
| Z => fun _ => a
| succ n => fun _ => s n ((Mod2_ind P a s mod') n)
end) mod'.
Axiom Mod2_ind_beta_mod : forall
(P : Mod2 -> Type)
(a : P Z)
(s : forall (n : Mod2), P n -> P (succ n))
(mod' : mod # a = s (succ Z) (s Z a))
, apD (Mod2_ind P a s mod') mod = mod'.
Fixpoint Mod2_rec
(P : Type)
(a : P)
(s : P -> P)
(mod' : a = s (s a))
(x : Mod2)
{struct x}
: P
:=
(match x return _ -> P with
| Z => fun _ => a
| succ n => fun _ => s ((Mod2_rec P a s mod') n)
end) mod'.
Axiom Mod2_rec_beta_mod : forall
(P : Type)
(a : P)
(s : P -> P)
(mod' : a = s (s a))
, ap (Mod2_rec P a s mod') mod = mod'.
End modulo.
Module Export moduloAlt.
Private Inductive Mod2A : Type0 :=
| ZA : Mod2A
| succA : Mod2A -> Mod2A.
Axiom modA : forall n : Mod2A, n = succA(succA n).
Fixpoint Mod2A_ind
(P : Mod2A -> Type)
(z : P ZA)
(s : forall n : Mod2A, P n -> P (succA n))
(mod' : forall (n : Mod2A) (a : P n),
modA n # a = s (succA n) (s n a))
(x : Mod2A)
{struct x}
: P x
:=
(match x return _ -> P x with
| ZA => fun _ => z
| succA n => fun _ => s n ((Mod2A_ind P z s mod') n)
end) mod'.
Axiom Mod2A_ind_beta_mod : forall
(P : Mod2A -> Type)
(z : P ZA)
(s : forall n : Mod2A, P n -> P (succA n))
(mod' : forall (n : Mod2A) (a : P n),
modA n # a = s (succA n) (s n a))
(n : Mod2A)
, apD (Mod2A_ind P z s mod') (modA n) = mod' n (Mod2A_ind P z s mod' n).
Fixpoint Mod2A_rec
(P : Type)
(z : P)
(s : P -> P)
(mod' : forall (a : P),
a = s (s a))
(x : Mod2A)
{struct x}
: P
:=
(match x return _ -> P with
| ZA => fun _ => z
| succA n => fun _ => s ((Mod2A_rec P z s mod') n)
end) mod'.
Axiom Mod2A_rec_beta_mod : forall
(P : Type)
(z : P)
(s : P -> P)
(mod' : forall (a : P),
a = s (s a))
(n : Mod2A)
, ap (Mod2A_rec P z s mod') (modA n) = mod' (Mod2A_rec P z s mod' n).
End moduloAlt.
Definition negate : Mod2 -> Mod2.
Proof.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
apply (succ Z).
Focus 2.
intros.
apply (succ H).
simpl.
rewrite transport_const.
rewrite <- mod.
reflexivity.
Defined.
Theorem modulo2 : forall n : Mod2, n = succ(succ n).
Proof.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
apply mod.
Focus 2.
intro n.
intro p.
apply (ap succ p).
simpl.
rewrite useful.
rewrite ap_idmap.
rewrite concat_Vp.
rewrite concat_1p.
rewrite ap_compose.
reflexivity.
Defined.
Definition plus : Mod2 -> Mod2 -> Mod2.
Proof.
intro n.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
apply n.
Focus 2.
intro m.
intro k.
apply (succ k).
simpl.
rewrite transport_const.
apply modulo2.
Defined.
Definition Bool_to_Mod2 : Bool -> Mod2.
Proof.
intro b.
destruct b.
apply (succ Z).
apply Z.
Defined.
Definition Mod2_to_Bool : Mod2 -> Bool.
Proof.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
apply false.
Focus 2.
intro n.
apply negb.
Focus 1.
simpl.
apply transport_const.
Defined.
Theorem eq1 : forall n : Bool, Mod2_to_Bool (Bool_to_Mod2 n) = n.
Proof.
intro b.
destruct b.
Focus 1.
compute.
reflexivity.
compute.
reflexivity.
Qed.
Theorem Bool_to_Mod2_negb : forall x : Bool,
succ (Bool_to_Mod2 x) = Bool_to_Mod2 (negb x).
Proof.
intros.
destruct x.
compute.
apply mod^.
compute.
apply reflexivity.
Defined.
Theorem eq2 : forall n : Mod2, Bool_to_Mod2 (Mod2_to_Bool n) = n.
Proof.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
compute.
reflexivity.
Focus 2.
intro n.
intro IHn.
symmetry.
transitivity (succ (Bool_to_Mod2 (Mod2_to_Bool n))).
Focus 1.
symmetry.
apply (ap succ IHn).
transitivity (Bool_to_Mod2 (negb (Mod2_to_Bool n))).
apply Bool_to_Mod2_negb.
enough (negb (Mod2_to_Bool n) = Mod2_to_Bool (succ n)).
apply (ap Bool_to_Mod2 X).
compute.
reflexivity.
simpl.
rewrite concat_p1.
rewrite concat_1p.
rewrite useful.
rewrite concat_p1.
rewrite ap_idmap.
rewrite ap_compose.
enough (ap Mod2_to_Bool mod = reflexivity false).
rewrite X.
simpl.
rewrite concat_1p.
rewrite inv_V.
reflexivity.
enough (IsHSet Bool).
apply axiomK_hset.
apply X.
apply hset_bool.
Defined.
Theorem adj :
forall x : Mod2, eq1 (Mod2_to_Bool x) = ap Mod2_to_Bool (eq2 x).
Proof.
intro x.
enough (IsHSet Bool).
apply set_path2.
apply hset_bool.
Defined.
Definition isomorphism : IsEquiv Mod2_to_Bool.
Proof.
apply (BuildIsEquiv Mod2 Bool Mod2_to_Bool Bool_to_Mod2 eq1 eq2 adj).
Qed.
Definition Mod2ToMod2A : Mod2 -> Mod2A.
Proof.
refine (Mod2_rec _ _ _ _).
Unshelve.
Focus 2.
apply ZA.
Focus 2.
apply succA.
Focus 1.
simpl.
apply modA.
Defined.
Definition Mod2AToMod2 : Mod2A -> Mod2.
Proof.
refine (Mod2A_rec _ _ _ _).
Unshelve.
Focus 1.
apply Z.
Focus 2.
apply succ.
Focus 1.
intro a.
apply (modulo2 a).
Defined.
Lemma Mod2AToMod2succA :
forall (n : Mod2A), Mod2AToMod2(succA n) = succ (Mod2AToMod2 n).
Proof.
reflexivity.
Defined.
Lemma Mod2ToMod2Asucc :
forall (n : Mod2), Mod2ToMod2A(succ n) = succA (Mod2ToMod2A n).
Proof.
reflexivity.
Defined.
Theorem eqI1 : forall (n : Mod2), n = Mod2AToMod2(Mod2ToMod2A n).
Proof.
refine (Mod2_ind _ _ _ _).
Unshelve.
Focus 2.
reflexivity.
Focus 2.
intro n.
intro H.
rewrite Mod2ToMod2Asucc.
rewrite Mod2AToMod2succA.
rewrite <- H.
reflexivity.
simpl.
rewrite useful.
rewrite ap_idmap.
rewrite concat_p1.
rewrite ap_compose.
rewrite Mod2_rec_beta_mod.
rewrite Mod2A_rec_beta_mod.
simpl.
simpl.
enough (modulo2 Z = mod).
rewrite X.
apply concat_Vp.
compute.
reflexivity.
Defined.
Theorem eqI2 : forall (n : Mod2A), n = Mod2ToMod2A(Mod2AToMod2 n).
Proof.
refine (Mod2A_ind _ _ _ _).
Focus 1.
reflexivity.
Unshelve.
Focus 2.
intros.
rewrite Mod2AToMod2succA.
rewrite Mod2ToMod2Asucc.
rewrite <- X.
reflexivity.
intros.
simpl.
rewrite useful.
rewrite ap_idmap.
rewrite ap_compose.
rewrite Mod2A_rec_beta_mod.