mirror of https://github.com/nmvdw/HITs-Examples
Added old proofs in new style (missing: Z' is hset)
This commit is contained in:
Niels
parent
8cf115c9ab
commit
efec2e88f8
699
int/Integers.v
699
int/Integers.v
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@ -76,7 +76,7 @@ Module Export Ints.
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End Z_recursion.
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Instance FSet_recursion : HitRecursion Z :=
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Instance Z_recursion : HitRecursion Z :=
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{
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indTy := _; recTy := _;
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H_inductor := Z_ind; H_recursor := Z_rec }.
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@ -477,9 +477,6 @@ End initial_Z.
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(*
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Module Export AltInts.
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Private Inductive Z' : Type0 :=
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@ -488,443 +485,263 @@ Module Export AltInts.
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Axiom path : positive 0 = negative 0.
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Fixpoint Z'_ind
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(P : Z' -> Type)
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(po : forall (x : nat), P (positive x))
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(ne : forall (x : nat), P (negative x))
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(i : path # (po 0) = ne 0)
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(x : Z')
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{struct x}
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: P x
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:=
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(match x return _ -> P x with
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| positive n => fun _ => po n
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| negative n => fun _ => ne n
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end) i.
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Section AltIntsInd.
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Variable (P : Z' -> Type)
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(po : forall (x : nat), P (positive x))
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(ne : forall (x : nat), P (negative x))
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(i : path # (po 0) = ne 0).
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Axiom Z'_ind_beta_inv1 : forall
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(P : Z' -> Type)
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(po : forall (x : nat), P (positive x))
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(ne : forall (x : nat), P (negative x))
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(i : path # (po 0) = ne 0)
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, apD (Z'_ind P po ne i) path = i.
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Fixpoint Z'_ind (x : Z') {struct x} : P x
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:=
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(match x return _ -> P x with
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| positive n => fun _ => po n
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| negative n => fun _ => ne n
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end) i.
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Axiom Z'_ind_beta_path : apD Z'_ind path = i.
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End AltIntsInd.
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Section AltIntsRec.
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Variable (P : Type)
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(po : nat -> P)
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(pn : nat -> P)
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(i : po 0 = pn 0).
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Definition Z'_rec : Z' -> P.
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Proof.
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simple refine (Z'_ind _ _ _ _) ; simpl.
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- apply po.
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- apply pn.
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- refine (transport_const _ _ @ i).
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Defined.
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Definition Z'_rec_beta_path : ap Z'_rec path = i.
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Proof.
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unfold Z_rec.
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eapply (cancelL (transport_const path _)).
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simple refine ((apD_const _ _)^ @ _).
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apply Z'_ind_beta_path.
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Defined.
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End AltIntsRec.
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Instance Z'_recursion : HitRecursion Z' :=
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{
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indTy := _; recTy := _;
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H_inductor := Z'_ind; H_recursor := Z'_rec
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}.
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End AltInts.
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Definition succ_Z' : Z' -> Z'.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intro n.
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apply (positive (S n)).
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Focus 2.
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intro n.
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induction n.
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apply (positive 1).
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apply (negative n).
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simpl.
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rewrite transport_const.
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reflexivity.
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Defined.
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Definition pred_Z' : Z' -> Z'.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intro n.
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induction n.
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apply (negative 1).
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apply (positive n).
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Focus 2.
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intro n.
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apply (negative (S n)).
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simpl.
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rewrite transport_const.
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reflexivity.
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Defined.
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Fixpoint Nat_to_Pos (n : nat) : Pos :=
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match n with
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| 0 => Int.one
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| S k => succ_pos (Nat_to_Pos k)
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end.
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Definition Z'_to_Int : Z' -> Int.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intro x.
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induction x.
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apply (Int.zero).
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apply (succ_int IHx).
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Focus 2.
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intro x.
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induction x.
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apply (Int.zero).
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apply (pred_int IHx).
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simpl.
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rewrite transport_const.
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reflexivity.
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Defined.
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Definition Pos_to_Nat : Pos -> nat.
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Proof.
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intro x.
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induction x.
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apply 1.
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apply (S IHx).
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Defined.
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Definition Int_to_Z' (x : Int) : Z'.
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Proof.
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induction x.
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apply (negative (Pos_to_Nat p)).
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apply (positive 0).
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apply (positive (Pos_to_Nat p)).
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Defined.
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Lemma Z'_to_int_pos_homomorphism :
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forall n : nat, Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)).
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Proof.
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intro n.
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reflexivity.
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Qed.
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Lemma Z'_to_int_neg_homomorphism :
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forall n : nat, Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)).
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Proof.
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intro n.
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reflexivity.
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Qed.
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Theorem isoEq1 : forall x : Int, Z'_to_Int(Int_to_Z' x) = x.
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Proof.
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intro x.
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induction x.
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induction p.
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reflexivity.
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rewrite Z'_to_int_neg_homomorphism.
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rewrite IHp.
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reflexivity.
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reflexivity.
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induction p.
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reflexivity.
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rewrite Z'_to_int_pos_homomorphism.
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rewrite IHp.
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reflexivity.
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Defined.
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Lemma Int_to_Z'_succ_homomorphism :
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forall x : Int, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x).
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Proof.
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simpl.
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intro x.
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simpl.
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induction x.
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induction p.
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compute.
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apply path.
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reflexivity.
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reflexivity.
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induction p.
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reflexivity.
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reflexivity.
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Qed.
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Lemma Int_to_Z'_pred_homomorphism :
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forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x).
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Proof.
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intro x.
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induction x.
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induction p.
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reflexivity.
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reflexivity.
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reflexivity.
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induction p.
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reflexivity.
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reflexivity.
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Qed.
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Theorem isoEq2 : forall x : Z', Int_to_Z'(Z'_to_Int x) = x.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intro x.
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induction x.
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reflexivity.
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rewrite Z'_to_int_pos_homomorphism.
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rewrite Int_to_Z'_succ_homomorphism.
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rewrite IHx.
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reflexivity.
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Focus 2.
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intro x.
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induction x.
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apply path.
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rewrite Z'_to_int_neg_homomorphism.
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rewrite Int_to_Z'_pred_homomorphism.
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rewrite IHx.
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reflexivity.
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simpl.
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rewrite @HoTT.Types.Paths.transport_paths_FlFr.
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rewrite concat_p1.
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rewrite ap_idmap.
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enough (ap (fun x : Z' => Z'_to_Int x) path = reflexivity Int.zero).
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rewrite ap_compose.
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rewrite X.
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apply concat_1p.
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apply axiomK_hset.
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apply hset_int.
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Defined.
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Theorem adj :
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forall x : Z', isoEq1 (Z'_to_Int x) = ap Z'_to_Int (isoEq2 x).
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Proof.
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intro x.
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apply hset_int.
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Defined.
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Definition isomorphism : IsEquiv Z'_to_Int.
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Proof.
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apply (BuildIsEquiv Z' Int Z'_to_Int Int_to_Z' isoEq1 isoEq2 adj).
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Qed.
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Axiom everythingSet : forall T : Type, IsHSet T.
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Definition Z_to_Z' : Z -> Z'.
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Proof.
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refine (Z_rec _ _ _ _ _ _).
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Unshelve.
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Focus 1.
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apply (positive 0).
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Focus 3.
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apply succ_Z'.
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Focus 3.
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apply pred_Z'.
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Focus 1.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intros.
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reflexivity.
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Focus 2.
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intros.
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induction x.
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Focus 1.
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compute.
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apply path^.
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reflexivity.
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apply everythingSet.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intros.
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induction x.
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Focus 1.
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compute.
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apply path.
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reflexivity.
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Focus 2.
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intros.
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reflexivity.
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apply everythingSet.
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Defined.
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Definition Z'_to_Z : Z' -> Z.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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induction 1.
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apply nul.
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apply (succ IHx).
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Focus 2.
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induction 1.
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Focus 1.
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apply nul.
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apply (pred IHx).
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simpl.
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rewrite transport_const.
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reflexivity.
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Defined.
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Theorem isoZEq1 : forall n : Z', Z_to_Z'(Z'_to_Z n) = n.
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Proof.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 3.
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intros.
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induction x.
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compute.
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apply path.
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transitivity (Z_to_Z' (pred (Z'_to_Z (negative x)))).
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enough (Z'_to_Z (negative x.+1) = pred (Z'_to_Z (negative x))).
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rewrite X.
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reflexivity.
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reflexivity.
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transitivity (pred_Z' (Z_to_Z' (Z'_to_Z (negative x)))).
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Focus 1.
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reflexivity.
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rewrite IHx.
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reflexivity.
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Focus 2.
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intros.
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induction x.
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Focus 1.
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reflexivity.
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transitivity (Z_to_Z' (succ (Z'_to_Z (positive x)))).
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reflexivity.
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transitivity (succ_Z' (Z_to_Z' (Z'_to_Z (positive x)))).
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reflexivity.
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rewrite IHx.
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reflexivity.
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apply everythingSet.
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Defined.
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Theorem isoZEq2 : forall n : Z, Z'_to_Z(Z_to_Z' n) = n.
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Proof.
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refine (Z_ind _ _ _ _ _ _).
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Unshelve.
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Focus 1.
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reflexivity.
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Focus 1.
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intros.
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apply everythingSet.
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Focus 1.
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intros.
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apply everythingSet.
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Focus 1.
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intro n.
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intro X.
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transitivity (Z'_to_Z (succ_Z' (Z_to_Z' n))).
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reflexivity.
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transitivity (succ (Z'_to_Z (Z_to_Z' n))).
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Focus 2.
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rewrite X.
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reflexivity.
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enough (forall m : Z', Z'_to_Z (succ_Z' m) = succ (Z'_to_Z m)).
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rewrite X0.
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reflexivity.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 2.
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intros.
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reflexivity.
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Focus 2.
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intros.
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induction x.
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Focus 1.
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reflexivity.
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compute.
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rewrite <- inv2.
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reflexivity.
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apply everythingSet.
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intros.
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transitivity (Z'_to_Z (pred_Z' (Z_to_Z' n))).
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reflexivity.
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transitivity (pred (Z'_to_Z (Z_to_Z' n))).
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Focus 2.
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rewrite X.
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reflexivity.
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enough (forall m : Z', Z'_to_Z (pred_Z' m) = pred (Z'_to_Z m)).
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rewrite X0.
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reflexivity.
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refine (Z'_ind _ _ _ _).
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Unshelve.
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Focus 1.
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apply everythingSet.
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Focus 1.
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intro x.
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induction x.
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reflexivity.
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compute.
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rewrite <- inv1.
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reflexivity.
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intro x.
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reflexivity.
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Defined.
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Theorem adj2 :
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forall x : Z', isoZEq2 (Z'_to_Z x) = ap Z'_to_Z (isoZEq1 x).
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Proof.
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intro x.
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apply everythingSet.
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Defined.
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Definition isomorphism2 : IsEquiv Z'_to_Z.
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Proof.
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apply (BuildIsEquiv Z' Z Z'_to_Z Z_to_Z' isoZEq2 isoZEq1 adj2).
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Qed.
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*)
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Section Isomorphic.
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Definition succ_Z' : Z' -> Z'.
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Proof.
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simple refine (Z'_rec _ _ _ _).
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- apply (fun n => positive (S n)).
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- induction 1 as [ | n].
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* apply (positive 1).
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* apply (negative n).
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- reflexivity.
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Defined.
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Definition pred_Z' : Z' -> Z'.
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Proof.
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simple refine (Z'_rec _ _ _ _).
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- induction 1 as [ | n].
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* apply (negative 1).
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* apply (positive n).
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- intro n.
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apply (negative (S n)).
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- reflexivity.
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Defined.
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Fixpoint Nat_to_Pos (n : nat) : Int.Pos :=
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match n with
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| 0 => Int.one
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| S k => succ_pos (Nat_to_Pos k)
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end.
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Definition Z'_to_Int : Z' -> Int.
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Proof.
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simple refine (Z'_rec _ _ _ _).
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- induction 1 as [ | n IHn].
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apply (Int.zero).
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apply (succ_int IHn).
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- induction 1 as [ | n IHn].
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apply (Int.zero).
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apply (pred_int IHn).
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- reflexivity.
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Defined.
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Definition Pos_to_Nat : Int.Pos -> nat.
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Proof.
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induction 1 as [ | n IHn].
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- apply 1.
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- apply (S IHn).
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Defined.
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Definition Int_to_Z' (x : Int) : Z'.
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Proof.
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induction x as [p | | p].
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apply (negative (Pos_to_Nat p)).
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apply (positive 0).
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apply (positive (Pos_to_Nat p)).
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Defined.
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Definition Z'_to_int_pos_homomorphism n :
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Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)) := idpath.
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Definition Z'_to_int_neg_homomorphism n :
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Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)) := idpath.
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Theorem Int_to_Z'_to_Int : forall x : Int, Z'_to_Int(Int_to_Z' x) = x.
|
||||
Proof.
|
||||
induction x as [p | | p].
|
||||
- induction p as [ | p IHp ].
|
||||
* reflexivity.
|
||||
* refine (Z'_to_int_neg_homomorphism _ @ ap pred_int IHp).
|
||||
- reflexivity.
|
||||
- induction p as [ | p IHp ].
|
||||
* reflexivity.
|
||||
* refine (Z'_to_int_pos_homomorphism _ @ ap succ_int IHp).
|
||||
Defined.
|
||||
|
||||
Lemma Int_to_Z'_succ_homomorphism :
|
||||
forall x, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x).
|
||||
Proof.
|
||||
induction x as [p | | p].
|
||||
- induction p as [ | p IHp] ; cbn.
|
||||
* apply path.
|
||||
* reflexivity.
|
||||
- reflexivity.
|
||||
- induction p ; reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma Int_to_Z'_pred_homomorphism :
|
||||
forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x).
|
||||
Proof.
|
||||
induction x as [p | | p] ; try (induction p) ; reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem Z'_to_Int_to_Z' : forall x : Z', Int_to_Z'(Z'_to_Int x) = x.
|
||||
Proof.
|
||||
simple refine (Z'_ind _ _ _ _) ; simpl.
|
||||
- induction x as [ | x] ; simpl.
|
||||
* reflexivity.
|
||||
* refine (ap Int_to_Z' (Z'_to_int_pos_homomorphism _)^ @ _) ; simpl.
|
||||
refine (Int_to_Z'_succ_homomorphism _ @ _).
|
||||
apply (ap succ_Z' IHx).
|
||||
- induction x as [ | x IHx] ; simpl.
|
||||
* apply path.
|
||||
* refine (ap Int_to_Z' (Z'_to_int_neg_homomorphism _)^ @ _) ; simpl.
|
||||
refine (Int_to_Z'_pred_homomorphism _ @ _).
|
||||
apply (ap pred_Z' IHx).
|
||||
- simpl.
|
||||
refine (transport_paths_FlFr path _ @ _).
|
||||
refine (ap (fun z => _ @ z) (ap_idmap _) @ _).
|
||||
refine (ap (fun z => z @ _) (concat_p1 _) @ _).
|
||||
assert (ap (fun x : Z' => Z'_to_Int x) path = idpath) as X.
|
||||
{
|
||||
apply axiomK_hset ; apply hset_int.
|
||||
}
|
||||
refine (ap (fun z => z^ @ path) (ap_compose Z'_to_Int Int_to_Z' path) @ _).
|
||||
refine (ap (fun z => (ap Int_to_Z' z)^ @ _) X @ _).
|
||||
apply concat_1p.
|
||||
Defined.
|
||||
|
||||
Definition biinv_Z'_to_Int : BiInv Z'_to_Int :=
|
||||
((Int_to_Z' ; Z'_to_Int_to_Z'), (Int_to_Z' ; Int_to_Z'_to_Int)).
|
||||
|
||||
Definition equiv_Z'_to_Int : IsEquiv Z'_to_Int :=
|
||||
isequiv_biinv _ biinv_Z'_to_Int.
|
||||
|
||||
Instance Z'_set : IsHSet Z'.
|
||||
Proof.
|
||||
intros x y p q.
|
||||
|
||||
Admitted.
|
||||
|
||||
Definition Z_to_Z' : Z -> Z'.
|
||||
Proof.
|
||||
hrecursion.
|
||||
- apply (positive 0).
|
||||
- apply succ_Z'.
|
||||
- apply pred_Z'.
|
||||
- hinduction.
|
||||
* apply (fun _ => idpath).
|
||||
* induction x as [ | x IHx] ; simpl.
|
||||
** apply path^.
|
||||
** reflexivity.
|
||||
* apply set_path2.
|
||||
- hinduction.
|
||||
* induction x as [ | x IHx] ; simpl.
|
||||
** apply path.
|
||||
** reflexivity.
|
||||
* apply (fun _ => idpath).
|
||||
* apply set_path2.
|
||||
Defined.
|
||||
|
||||
Definition Z'_to_Z : Z' -> Z.
|
||||
Proof.
|
||||
hrecursion.
|
||||
- induction 1 as [ | x IHx].
|
||||
* apply zero_Z.
|
||||
* apply (succ IHx).
|
||||
- induction 1 as [ | x IHx].
|
||||
* apply zero_Z.
|
||||
* apply (pred IHx).
|
||||
- reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem Z'_to_Z_to_Z' : forall n : Z', Z_to_Z'(Z'_to_Z n) = n.
|
||||
Proof.
|
||||
hinduction.
|
||||
- induction x as [ | x IHx] ; cbn.
|
||||
* reflexivity.
|
||||
* apply (ap succ_Z' IHx).
|
||||
- induction x as [ | x IHx] ; cbn.
|
||||
* apply path.
|
||||
* apply (ap pred_Z' IHx).
|
||||
- apply set_path2.
|
||||
Defined.
|
||||
|
||||
Lemma Z'_to_Z_succ : forall n, Z'_to_Z(succ_Z' n) = succ(Z'_to_Z n).
|
||||
Proof.
|
||||
hinduction.
|
||||
- apply (fun _ => idpath).
|
||||
- induction x.
|
||||
* reflexivity.
|
||||
* apply inv2.
|
||||
- apply set_path2.
|
||||
Defined.
|
||||
|
||||
Lemma Z'_to_Z_pred : forall n, Z'_to_Z(pred_Z' n) = pred(Z'_to_Z n).
|
||||
Proof.
|
||||
hinduction.
|
||||
- induction x.
|
||||
* reflexivity.
|
||||
* apply inv1.
|
||||
- apply (fun _ => idpath).
|
||||
- apply set_path2.
|
||||
Defined.
|
||||
|
||||
Theorem Z_to_Z'_to_Z : forall n : Z, Z'_to_Z(Z_to_Z' n) = n.
|
||||
Proof.
|
||||
hinduction ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intros x Hx.
|
||||
refine (_ @ ap succ Hx).
|
||||
apply Z'_to_Z_succ.
|
||||
- intros x Hx.
|
||||
refine (_ @ ap pred Hx).
|
||||
apply Z'_to_Z_pred.
|
||||
Defined.
|
||||
|
||||
Definition biinv_Z'_to_Z : BiInv Z'_to_Z :=
|
||||
((Z_to_Z' ; Z'_to_Z_to_Z'), (Z_to_Z' ; Z_to_Z'_to_Z)).
|
||||
|
||||
Definition equiv_Z'_to_Z : IsEquiv Z'_to_Z :=
|
||||
isequiv_biinv _ biinv_Z'_to_Z.
|
||||
|
||||
End Isomorphic.
|
||||
|
|
|
|||
Loading…
Reference in New Issue