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Integers.v

@@ -1,25 +1,6 @@
 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 :=
@@ -175,7 +156,7 @@ refine (Z_rec _ _ _ _ _ _).
   Focus 2.
   apply loop.
 
-  rewrite useful.
+  rewrite @HoTT.Types.Paths.transport_paths_FlFr.
   rewrite ap_idmap.
   rewrite concat_Vp.
   rewrite concat_1p.
@@ -417,7 +398,7 @@ refine (Z'_ind _ _ _ _).
   reflexivity.
 
   simpl.
-  rewrite useful.
+  rewrite @HoTT.Types.Paths.transport_paths_FlFr.
   rewrite concat_p1.
   rewrite ap_idmap.
 

Mod2.v

@@ -1,25 +1,6 @@
 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 :=
@@ -72,65 +53,6 @@ Axiom Mod2_rec_beta_mod : forall
 
 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 _ _ _ _).
@@ -161,7 +83,7 @@ refine (Mod2_ind _ _ _ _).
  apply (ap succ p).
 
  simpl.
- rewrite useful.
+ rewrite @HoTT.Types.Paths.transport_paths_FlFr.
  rewrite ap_idmap.
  rewrite concat_Vp.
  rewrite concat_1p.
@@ -265,7 +187,7 @@ refine (Mod2_ind _ _ _ _).
     simpl.
     rewrite concat_p1.
     rewrite concat_1p.
-    rewrite useful.
+    rewrite @HoTT.Types.Paths.transport_paths_FlFr.
     rewrite concat_p1.
     rewrite ap_idmap.
     rewrite ap_compose.
@@ -296,101 +218,3 @@ 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.