(** This file contain various results that could be upstreamed to Foundations/PartA.v *)
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.Tactics.

(** * Generalisations of [maponpaths]

The following are a uniformly-named set of lemmas giving how multi-argument (non-dependent) functions act on paths, generalising [maponpaths].

  The naming convention is that e.g. [maponpaths_135] takes paths in the 1st, 3rd, and 5th arguments, counting _backwards_ from the end.  (The “counting backwards” is so that it doesn’t depend on the total number of arguments the function takes.)

  All are defined in terms of [maponpaths], to allow use of lemmas about it for reasoning about these.

 See below for a note about defining duplicates/special cases of these lemmas downstream. *)

Definition maponpaths_1 {X A : UU} (f : X -> A) {x x'} (e : x = x')
  : f x = f x'
:= maponpaths f e.


Definition maponpaths_2 {Y X A : UU} (f : Y -> X -> A)
    {y y'} (e_y : y = y') x
  : f y x = f y' x
:= maponpaths (fun y => f y x) e_y.

(* TODO: should this be defined in terms of [two_arg_paths] or [map_on_two_paths], from [Foundations]?*)
Definition maponpaths_12 {Y X A : UU} (f : Y -> X -> A)
    {y y'} (e_y : y = y') {x x'} (e_x : x = x')
  : f y x = f y' x'
:= maponpaths_1 _ e_x @ maponpaths_2 f e_y _.

Definition maponpaths_3 {Z Y X A : UU} (f : Z -> Y -> X -> A)
    {z z'} (e_z : z = z') y x
  : f z y x = f z' y x
:= maponpaths (fun z => f z y x) e_z.

Definition maponpaths_123 {Z Y X A : UU} (f : Z -> Y -> X -> A)
    {z z'} (e_z : z = z') {y y'} (e_y : y = y') {x x'} (e_x : x = x')
  : f z y x = f z' y' x'
:= maponpaths_12 _ e_y e_x @ maponpaths_3 f e_z _ _.

Definition maponpaths_13 {Z Y X A : UU} (f : Z -> Y -> X -> A)
    {z z'} (e_z : z = z') (y:Y) {x x'} (e_x : x = x')
  : f z y x = f z' y x'
:= maponpaths_123 _ e_z (idpath y) e_x.

Definition maponpaths_4 {W Z Y X A : UU} (f : W -> Z -> Y -> X -> A)
    {w w'} (e_w : w = w') z y x
  : f w z y x = f w' z y x
:= maponpaths (fun w => f w z y x) e_w.

Definition maponpaths_1234 {W Z Y X A : UU} (f : W -> Z -> Y -> X -> A)
    {w w'} (e_w : w = w') {z z'} (e_z : z = z')
    {y y'} (e_y : y = y') {x x'} (e_x : x = x')
  : f w z y x = f w' z' y' x'
:= maponpaths_123 _ e_z e_y e_x @ maponpaths_4 _ e_w _ _ _.

(** Notes on duplication issues:

Duplicates and special cases of these lemmas have been re-written multiple times, in multiple packages, by multiple contributors.

Sometimes this is intended and useful — e.g. for frequently-used specialisations such as [base_paths], especially when subsequent definitions mention them, like [fiber_paths].

Other times, it just happens because the contributor hasn’t been aware of the applicable general versions (or they weren’t available at the time of writing).  These cases are code duplication, causing redundancy and inconsistent coding style, and should be avoided/refactored.

To keep these clearly distinguished: if you are defining duplicates or special cases of these lemmas, please (a) define them in terms of the general ones, so that lemmas about these are applicable, and (b) add a comment noting why your definitions are desirable.

A few direct duplicates remain:

- [Foundations.PartA.two_arg_paths] and [Foundations.PartA.map_on_two_paths] are complete duplicates of [maponpaths_12], and of each other (modulo reordering of arguments).
- [Foundations.PartA.pathsdirprod] and [Foundations.PartA.total2_paths2] are complete duplicates of each other.
 *)

(* TODO: several of the [maponpaths] family could usefully be generalised, to work with functions whose output type is dependent on the arguments that aren’t being varied in the specific lemma. *)

Lemma maponpaths_for_constant_function {T1 T2 : UU} (x : T2) {t1 t2 : T1}
      (e: t1 = t2): maponpaths (fun _: T1 => x) e = idpath x.
Proof.
  induction e.
  apply idpath.
Qed.

Lemma base_paths_pair_path_in2 {X : UU} (P : X → UU) {x : X} {p q : P x} (e : p = q) :
  base_paths _ _ (pair_path_in2 P e) = idpath x.
Proof.
now induction e.
Qed.

Lemma transportf_transpose_right
      {X : UU}
      {P : X → UU}
      {x x' : X}
      {e : x = x'}
      {y : P x}
      {y' : P x'} :
      transportb P e y' = y -> y' = transportf P e y.
Proof.
  induction e; apply idfun.
Defined.

Definition transportb_transpose_right
           {A : UU}
           {P : A → UU}
           {x x' : A}
           {e : x = x'}
           {y : P x} {y' : P x'}
  : transportf P e y = y' → y = transportb P e y'.
Proof.
  induction e; apply idfun.
Defined.

Definition transportf_transpose_left
           {X : UU}
           {P : X → UU}
           {x x' : X}
           {e : x = x'}
           {y : P x}
           {y' : P x'}
  : y = transportb P e y' → transportf P e y = y'.
Proof.
  induction e; apply idfun.
Defined.

Definition transportb_transpose_left
           {X : UU}
           {P : X → UU}
           {x x' : X}
           {e : x = x'}
           {y : P x}
           {y' : P x'}
  : y' = transportf P e y → transportb P e y' = y.
Proof.
  induction e; apply idfun.
Defined.

(* taken from TypeTheory/Display_Cats/Auxiliary.v *)
(** Very handy for reasoning with “dependent paths” —

Note: similar to [transportf_pathsinv0_var], [transportf_pathsinv0'],
but not quite a special case of them, or (as far as I can find) any other
library lemma.
 *)

Definition transportf_pathsinv0 {X} (P:X->UU) {x y:X} (p:x = y) (u:P x) (v:P y) :
  transportf _ (!p) v = u -> transportf _ p u = v.
Proof. intro e. induction p, e. reflexivity. Defined.

Lemma transportf_comp_lemma (X : UU) (B : X -> UU) {A A' A'': X} (e : A = A'') (e' : A' = A'')
  (x : B A) (x' : B A')
  : transportf _ (e @ !e') x = x'
  -> transportf _ e x = transportf _ e' x'.
Proof.
  intro H.
  eapply pathscomp0.
  2: { apply maponpaths. exact H. }
  eapply pathscomp0.
  2: { symmetry. apply transport_f_f. }
  apply (maponpaths (λ p, transportf _ p x)).
  apply pathsinv0.
  eapply pathscomp0.
  - apply @pathsinv0, path_assoc.
  - eapply pathscomp0.
    apply maponpaths.
    apply pathsinv0l.
    apply pathscomp0rid.
Defined.

Lemma transportf_comp_lemma_hset (X : UU) (B : X -> UU) (A : X) (e : A = A)
  {x x' : B A} (hs : isaset X)
  : x = x'
  -> transportf _ e x = x'.
Proof.
  intros ex.
  apply @pathscomp0 with (transportf _ (idpath _) x).
  - apply (maponpaths (λ p, transportf _ p x)).
    apply hs.
  - exact ex.
Qed.


Lemma transportf_bind {X : UU} {P : X → UU}
  {x x' x'' : X} (e : x' = x) (e' : x = x'')
  y y'
: y = transportf P e y' -> transportf _ e' y = transportf _ (e @ e') y'.
Proof.
  intro H; destruct e, e'; exact H.
Defined.

Lemma pathscomp0_dep {X : UU} {P : X → UU}
  {x x' x'' : X} {e : x' = x} {e' : x'' = x'}
  {y} {y'} {y''}
: (y = transportf P e y') -> (y' = transportf _ e' y'')
  -> y = transportf _ (e' @ e) y''.
Proof.
  intros ee ee'.
  etrans. apply ee.
  apply transportf_bind, ee'.
Defined.

Tactic Notation "etrans_dep" := eapply @pathscomp0_dep.



Lemma transportf_set {A : UU} (B : A → UU)
      {a : A} (e : a = a) (b : B a)
      (X : isaset A)
  : transportf B e b = b.
Proof.
  apply transportf_comp_lemma_hset.
  - apply X.
  - apply idpath.
Defined.


Lemma transportf_pair {A B} (P : A × B -> UU) {a a' : A} {b b' : B}
      (eA : a = a') (eB : b = b') (p : P (a,,b))
      : transportf P (pathsdirprod eA eB) p =
        transportf (λ bb, P(a',,bb) ) eB (transportf (λ aa, P(aa,,b)) eA p).
Proof.
  induction eA. induction eB. apply idpath.
Defined.

Lemma weqhomot {A B : UU} (f : A -> B) (w : A ≃ B) (H : w ~ f) : isweq f.
Proof.
  apply isweqhomot with w. apply H. apply weqproperty.
Defined.

Lemma invmap_eq {A B : UU} (f : A ≃ B) (b : B) (a : A)
  : b = f a → invmap f b = a.
Proof.
  intro H.
  apply (invmaponpathsweq f).
  etrans. apply homotweqinvweq. apply H.
Defined.

Lemma pr1_transportf {A : UU} {B : A -> UU} {P : ∏ a, B a -> UU}
   {a a' : A} (e : a = a') (xs : ∑ b : B a, P _ b):
   pr1 (transportf (λ x, ∑ b : B x, P _ b) e xs) =
     transportf (λ x, B x) e (pr1 xs).
Proof.
  apply pathsinv0.
  apply (transport_map (λ a, pr1 (P := P a))).
Defined.

Lemma pr1_transportb {A : UU} {B : A -> UU} {P : ∏ a, B a -> UU}
   {a a' : A} (e : a = a') (xs : ∑ b : B a', P _ b):
   pr1 (transportb (λ x, ∑ b : B x, P _ b) e xs) =
     transportb (λ x, B x) e (pr1 xs).
Proof.
  apply pathsinv0.
  apply (transport_map (λ a, pr1 (P := P a))).
Defined.

Lemma pr2_transportf {A} {B1 B2 : A → UU}
    {a a' : A} (e : a = a') (xs : B1 a × B2 a)
  : pr2 (transportf (λ a, B1 a × B2 a) e xs) = transportf _ e (pr2 xs).
Proof.
  apply pathsinv0.
  apply (transport_map (λ a, pr2 (P := λ _, B2 a))).
Defined.

Lemma pr2_transportb {A} {B1 B2 : A → UU}
    {a a' : A} (e : a = a') (xs : B1 a' × B2 a')
  : pr2 (transportb (λ a, B1 a × B2 a) e xs) = transportb _ e (pr2 xs).
Proof.
  apply pathsinv0.
  apply (transport_map (λ a, pr2 (P := λ _, B2 a))).
Defined.

Lemma coprodcomm_coprodcomm {X Y : UU} (v : X ⨿ Y) : coprodcomm Y X (coprodcomm X Y v) = v.
Proof.
  induction v as [x|y]; reflexivity.
Defined.

Definition sumofmaps_funcomp {X1 X2 Y1 Y2 Z : UU} (f1 : X1 → X2) (f2 : X2 → Z) (g1 : Y1 → Y2)
  (g2 : Y2 → Z) : sumofmaps (f2 ∘ f1) (g2 ∘ g1) ~ sumofmaps f2 g2 ∘ coprodf f1 g1.
Proof.
  intro x. induction x as [x|y]; reflexivity.
Defined.

Definition sumofmaps_homot {X Y Z : UU} {f f' : X → Z} {g g' : Y → Z} (h : f ~ f') (h2 : g ~ g')
  : sumofmaps f g ~ sumofmaps f' g'.
Proof.
  intro x. induction x as [x|y].
  - exact (h x).
  - exact (h2 y).
Defined.

(** coprod computation helper lemmas  *)

Definition coprod_rect_compute_1
           (A B : UU) (P : A ⨿ B -> UU)
           (f : ∏ (a : A), P (ii1 a))
           (g : ∏ (b : B), P (ii2 b)) (a:A) :
  coprod_rect P f g (ii1 a) = f a.
Proof.
  intros.
  apply idpath.
Defined.

Definition coprod_rect_compute_2
           (A B : UU) (P : A ⨿ B -> UU)
           (f : ∏ a : A, P (ii1 a))
           (g : ∏ b : B, P (ii2 b)) (b:B) :
  coprod_rect P f g (ii2 b) = g b.
Proof.
  intros.
  apply idpath.
Defined.

(** Flip the arguments of a function *)
Definition flipsec {A B : UU} {C : A -> B -> UU} (f : ∏ a b, C a b) : ∏ b a, C a b :=
  λ x y, f y x.
Notation flip := flipsec.

(** Flip is a weak equivalence (in fact, it is an involution) *)
Lemma isweq_flipsec {A B : UU} {C : A -> B -> UU} : isweq (@flipsec A B C).
Proof.
  apply (isweq_iso _ flipsec); reflexivity.
Defined.

Definition flipsec_weq {A B : UU} {C : A -> B -> UU} :
  (∏ a b, C a b) ≃ (∏ b a, C a b) := make_weq flipsec isweq_flipsec.

(** hlevel of empty type *)
Definition empty_hlevel
           (n : nat)
  : isofhlevel (n + 1) ∅.
Proof.
  induction n.
  - exact isapropempty.
  - exact (λ x, fromempty x).
Defined.

Definition empty_HLevel
           (n : nat)
  : HLevel (n + 1)
  := empty ,, empty_hlevel n.

Definition HLevel_fun
           {n : nat}
           (X Y : HLevel n)
  : HLevel n.
Proof.
  simple refine (_ ,, _).
  - exact (pr1 X → pr1 Y).
  - apply impredfun.
    exact (pr2 Y).
Defined.

(** The subtypes of a type of hlevel S n are also of hlevel S n.
    This doesn't work for types of hlevel 0: a subtype of a contractible
    type might be empty, not contractible! *)
Lemma isofhlevel_hsubtype {X : UU} {n : nat} (isof : isofhlevel (S n) X) :
  ∏ subt : hsubtype X, isofhlevel (S n) subt.
Proof.
  intros subt.
  apply isofhleveltotal2.
  - assumption.
  - intro.
    apply isofhlevelsnprop.
    apply propproperty.
Defined.

(** Homotopy equivalence on total spaces.

  This result combines weqfp and weqfibtototal conveniently.
  *)
Lemma weqtotal2 {X Y:Type} {P:X->Type} {Q:Y->Type} (f : X ≃ Y) :
  (∏ x, P x ≃ Q (f x)) -> (∑ x:X, P x) ≃ (∑ y:Y, Q y).
Proof.
  intros e. exists (λ xp, (f(pr1 xp),,e (pr1 xp) (pr2 xp))).
  exact (twooutof3c _ _ (isweqfibtototal P (Q ∘ f) e) (pr2 (weqfp f Q))).
Defined.

Lemma weq_subtypes'
    {X Y : UU} (w : X ≃ Y)
    {S : X -> UU} {T : Y -> UU}
    (HS : isPredicate S)
    (HT : isPredicate T)
    (HST : ∏ x : X, S x <-> T (w x))
  : (∑ x, S x) ≃ (∑ y, T y).
Proof.
  apply (weqbandf w).
  intros. apply weqiff.
  - apply HST.
  - apply HS.
  - apply HT.
Defined.

(* Specialisation of [weq_subtypes'] *)
Lemma weq_subtypes_iff
    {X : UU} {S T : X -> UU}
    (HS : isPredicate S)
    (HT : isPredicate T)
    (HST : ∏ x, S x <-> T x)
  : (∑ x, S x) ≃ (∑ x, T x).
Proof.
  apply (weq_subtypes' (idweq X)); assumption.
Defined.


Lemma hlevel_total2 n {A : UU} {B : A → UU} :
  isofhlevel n (∑ (x :A), B x) → isofhlevel (S n) A → ∏ (x : A), isofhlevel n (B x).
Proof.
  intros ic ia x.
  exact (isofhlevelweqf _ (invweq (ezweqpr1 _ _)) (isofhlevelffromXY _ _ ic ia _)).
Defined.

Definition path_sigma_hprop
           {A : UU}
           (B : A → UU)
           (x y : ∑ (z : A), B z)
           (HB : isaprop (B (pr1 y)))
  : x = y ≃ pr1 x = pr1 y.
Proof.
  refine (weqpr1 _ _ ∘ total2_paths_equiv _ _ _)%weq.
  intros.
  apply HB.
Defined.

Lemma iscontr_prod (X Y : UU)
  : iscontr X -> iscontr Y -> iscontr (X × Y).
Proof.
  exact (isofhleveldirprod 0 _ _).
Qed.

(** ** Pointed types *)

Section PointedTypes.
  Definition PointedType := ∑ X, X.
  Definition pointedType X x := X,,x : PointedType.
  Definition underlyingType (X:PointedType) := pr1 X.
  Coercion underlyingType : PointedType >-> UU.
  Definition basepoint (X:PointedType) := pr2 X.
  Definition loopSpace (X:PointedType) :=
    pointedType (basepoint X = basepoint X) (idpath _).
  Definition underlyingLoop {X:PointedType} (l:loopSpace X) : basepoint X = basepoint X.
  Proof.
    intros. exact l.
  Defined.
  Definition Ω := loopSpace.
End PointedTypes.

(** associativity of ∑ *)

Definition weq_total2_prod {X Y} (Z:Y->Type) : (∑ y, X × Z y) ≃ (X × ∑ y, Z y).
Proof.
  (* move upstream *)
  intros. simple refine (make_weq _ (isweq_iso _ _ _ _)).
  { intros [y [x z]]. exact (x,,y,,z). }
  { intros [x [y z]]. exact (y,,x,,z). }
  { intros [y [x z]]. reflexivity. }
  { intros [x [y z]]. reflexivity. }
Defined.

Definition totalAssociativity {X:UU} {Y: ∏ (x:X), UU} (Z: ∏ (x:X) (y:Y x), UU) : (∑ (x:X) (y:Y x), Z x y) ≃ (∑ (p:∑ (x:X), Y x), Z (pr1 p) (pr2 p)).
Proof.
  intros. simple refine (_,,isweq_iso _ _ _ _).
  { intros [x [y z]]. exact ((x,,y),,z). }
  { intros [[x y] z]. exact (x,,(y,,z)). }
  { intros [x [y z]]. reflexivity. }
  { intros [[x y] z]. reflexivity. }
Defined.

(* direct product of 3 paths; extends pathsdirprod *)
Definition paths3 {X Y Z} {x x':X} {y y':Y} {z z':Z} :
  x = x' -> y = y' -> z = z' -> @paths (_×_×_) (x,,y,,z) (x',,y',,z').
Proof.
  intros p q r. induction p, q, r. reflexivity.
Defined.

(* paths *)
Definition confun T {Y} (y:Y) := λ _:T, y.
Definition path_type {X} {x x':X} (p:x = x') := X.
Definition path_start {X} {x x':X} (p:x = x') := x.
Definition path_end {X} {x x':X} (p:x = x') := x'.

Definition uniqueness {T} (i:iscontr T) (t:T) : t = iscontrpr1 i.
Proof.
  intros. exact (pr2 i t).
Defined.

Definition uniqueness' {T} (i:iscontr T) (t:T) : iscontrpr1 i = t.
Proof.
  intros. exact (! (pr2 i t)).
Defined.

Definition path_inverse_to_right {X} {x y:X} (p q:x = y) : p = q -> !q@p = idpath _.
Proof.
  intros e. induction e. induction p. reflexivity.
Defined.

Definition path_inverse_to_right' {X} {x y:X} (p q:x = y) : p = q -> p@!q = idpath _.
Proof.
  intros e. induction e. induction p. reflexivity.
Defined.

(** ** Paths *)

Definition pathsinv0_to_right {X} {x y z:X} (p:y = x) (q:y = z) (r:x = z) :
  q = p @ r -> !p @ q = r.
Proof.
  intros e. induction p, q. exact e.
Defined.

Definition pathsinv0_to_right' {X} {x y:X} (p:y = x) (r:x = y) :
  idpath _ = p @ r -> !p = r.
Proof.
  intros e. induction p. exact e.
Defined.

Definition pathsinv0_to_right'' {X} {x:X} (p:x = x) :
  idpath _ = p -> !p = idpath _.
Proof.
  intros e. apply pathsinv0_to_right'. rewrite pathscomp0rid. exact e.
Defined.

Definition loop_power_nat {Y} {y:Y} (l:y = y) (n:nat) : y = y.
Proof.
  intros. induction n as [|n p].
  { exact (idpath _). }
  { exact (p@l). }
Defined.

Lemma irrel_paths {X} (irr:∏ x y:X, x = y) {x y:X} (p q:x = y) : p = q.
Proof.
  intros.
  assert (k : ∏ z:X, ∏ r:x = z, r @ irr z z = irr x z).
  { intros. induction r. reflexivity. }
  assert (l := k y p @ !k y q).
  apply (pathscomp_cancel_right _ _ _ l).
Defined.

Definition path_inv_rotate_2 {X} {a b c d:X} (p:a = b) (p':c = d) (q:a = c) (q':b = d) :
  q @ p' = p @ q' -> p' @ ! q' = !q @ p.
Proof.
  induction q,q'. simpl. repeat rewrite pathscomp0rid. apply idfun.
Defined.

Definition maponpaths_naturality {X Y : UU} {f : X -> Y}
           {x x' x'' : X} {p : x = x'} {q : x' = x''}
           {p': f x = f x'} {q': f x' = f x''}
           (r : maponpaths f p = p') (s : maponpaths f q = q') :
  maponpaths f (p @ q) = p' @ q'.
Proof.
  intros. induction r, s. apply maponpathscomp0.
Defined.

Definition maponpaths_naturality' {X Y : UU} {f : X -> Y}
           {x x' x'' : X} {p : x' = x} {q : x' = x''}
           {p' : f x' = f x} {q' : f x' = f x''}
           (r : maponpaths f p = p') (s : maponpaths f q = q') :
  maponpaths f (!p @ q) = (!p') @ q'.
Proof.
  intros. induction r, s, p, q. reflexivity.
Defined.


(** ** Pairs *)

Definition pr2_of_make_hfiber {X Y} {f:X->Y} {x:X} {y:Y} {e:f x = y} :
  pr2 (make_hfiber f x e) = e.
Proof.
  reflexivity.
Defined.

Definition pr2_of_pair {X} {P:X->Type} (x:X) (p:P x) : pr2 (tpair P x p) = p.
Proof.
  reflexivity.
Defined.

Definition pr2_of_make_weq {X Y} (f:X->Y) (i:isweq f) : pr2 (make_weq f i) = i.
Proof.
  reflexivity.
Defined.

(** ** Paths between pairs *)

(* replace all uses of this by uses of subtypePairEquality *)
Definition pair_path_props {X} {P:X->Type} {x y:X} {p:P x} {q:P y} :
  x = y -> (∏ z, isaprop (P z)) -> x,,p = y,,q.
Proof.
  intros e is. now apply subtypePairEquality.
Abort.

Local Open Scope transport.

Definition pair_path2 {A} {B:A->UU} {a a1 a2} {b1:B a1} {b2:B a2}
           (p:a1 = a) (q:a2 = a) (e:p#b1 = q#b2) : a1,,b1 = a2,,b2.
Proof.
  intros. induction p,q; compute in e. induction e. reflexivity.
Defined.

(** ** Projections from pair types *)

Definition pair_path_in2_comp1 {X} (P:X->Type) {x:X} {p q:P x} (e:p = q) :
  maponpaths pr1 (maponpaths (tpair P _) e) = idpath x.
Proof.
  intros. induction e. reflexivity.
Defined.

Definition total2_paths2_comp1 {X} {Y:X->Type} {x} {y:Y x} {x'} {y':Y x'}
           (p:x = x') (q:p#y = y') : maponpaths pr1 (two_arg_paths_f (f := tpair Y) p q) = p.
Proof.
  intros. induction p. induction q. reflexivity.
Defined.

Definition total2_paths2_comp2 {X} {Y:X->Type} {x} {y:Y x} {x'} {y':Y x'}
           (p:x = x') (q:p#y = y') :
  ! maponpaths_2 _ (total2_paths2_comp1 p q) y @ fiber_paths (two_arg_paths_f p q) = q.
Proof.
  intros. induction p, q. reflexivity.
Defined.

(** ** Maps from pair types *)

Definition from_total2 {X} {P:X->Type} {Y} : (∏ x, P x->Y) -> total2 P -> Y.
Proof.
  intros f [x p]. exact (f x p).
Defined.


(** ** Paths in coproducts *)

Definition inv_equality_by_case_equality_by_case
           {A B : UU}
           {x y : A ⨿ B}
           (p : x = y)
  : @inv_equality_by_case A B x y (equality_by_case p) = p.
Proof.
  induction x, y, p ; apply idpath.
Defined.

Definition equality_by_case_inv_equality_by_case
           {A B : UU}
           {x y : A ⨿ B}
           (p : equality_cases x y)
  : equality_by_case (inv_equality_by_case p) = p.
Proof.
  induction x, y, p; apply idpath.
Defined.

Lemma equality_by_case_equiv {A B} (t u : A ⨿ B) : (t = u) ≃ equality_cases t u.
Proof.
  use weq_iso.
  - apply equality_by_case.
  - apply inv_equality_by_case.
  - apply inv_equality_by_case_equality_by_case.
  - apply equality_by_case_inv_equality_by_case.
Defined.

Definition paths_inl_inl_equiv {A B : UU} (a a' : A)
  : @inl A B a = inl a' ≃ a = a'.
Proof.
  apply @equality_by_case_equiv.
Defined.

Definition paths_inl_inr_equiv {A B : UU} (a : A) (b : B)
  : inl a = inr b ≃ empty.
Proof.
  apply @equality_by_case_equiv.
Defined.

Definition paths_inr_inr_equiv {A B : UU} (b b' : B)
  : @inr A B b = inr b' ≃ b = b'.
Proof.
  apply @equality_by_case_equiv.
Defined.

Definition paths_inr_inl_equiv {A B : UU} (a : A) (b : B)
  : inr b = inl a ≃ empty.
Proof.
  apply @equality_by_case_equiv.
Defined.

(* Note: the following lemmas are near-duplicates of [isinclii1], [isinclii2] provided in [Foundations]. *)
Lemma isInjective_inl {A B : UU} : isInjective (@inl A B).
Proof.
  intros ? ?.
  refine (isweqinvmap (@paths_inl_inl_equiv _ _ _ _)).
Defined.

Lemma isInjective_inr {A B : UU} : isInjective (@inr A B).
Proof.
  intros ? ?.
  refine (isweqinvmap (@paths_inr_inr_equiv _ _ _ _)).
Defined.

(** ** Sections and functions *)

Notation homotsec := homot.

(** Naturality of homotopies *)
Definition homotsec_natural
           {X Y : UU}
           {f g : X → Y}
           (e : f ~ g)
           {x y : X}
           (p : x = y)
  : maponpaths f p @ e y = e x @ maponpaths g p.
Proof.
  induction p.
  exact (!(pathscomp0rid _)).
Defined.

(* compare with [adjev] *)
Definition evalat {T} {P:T->UU} (t:T) (f:∏ t:T, P t) := f t.

Definition apfun {X Y} {f f':X->Y} (p:f = f') {x x'} (q:x = x') : f x = f' x'.
  intros. induction q. exact (eqtohomot p x).
Defined.

Definition fromemptysec { X : empty -> UU } (nothing:empty) : X nothing.
(* compare with [fromempty] in u00 *)
Proof.
  induction nothing.
Defined.

Definition maponpaths_idpath {X Y} {f:X->Y} {x:X} : maponpaths f (idpath x) = idpath (f x).
Proof.
  intros. reflexivity.
Defined.

(** ** Transport *)

Definition cast {T U:Type} : T = U -> T -> U
  := transportf (λ T:Type, T).

Definition transport_type_path {X Y:Type} (p:X = Y) (x:X) :
  transportf (λ T:Type, T) p x = cast p x.
Proof.
  reflexivity.
Defined.

Definition transport_fun_path {X Y} {f g:X->Y} {x x':X} {p:x = x'} {e:f x = g x} {e':f x' = g x'} :
  e @ maponpaths g p = maponpaths f p @ e' ->
  transportf (λ x, f x = g x) p e = e'.
Proof.
  intros k. induction p. rewrite maponpaths_idpath in k. rewrite maponpaths_idpath in k.
  rewrite pathscomp0rid in k. exact k.
Defined.

Definition transportf_pathsinv0' {X} (P:X->UU) {x y:X} (p:x = y) (u:P x) (v:P y) :
  p # u = v -> !p # v = u.
Proof.
  intros e. induction p, e. reflexivity.
Defined.

Lemma transport_idfun {X} (P:X->UU) {x y:X} (p:x = y) (u:P x) :
  transportf P p u = transportf (idfun UU) (maponpaths P p) u.
(* same as HoTT.PathGroupoids.transport_idmap_ap *)
Proof.
  intros. induction p. reflexivity.
Defined.

Lemma transport_functions {X} {Y:X->Type} {Z:∏ x (y:Y x), Type}
      {f f':∏ x : X, Y x} (p:f = f') (z:∏ x, Z x (f x)) x :
    transportf (λ f, ∏ x, Z x (f x)) p z x =
    transportf (Z x) (toforallpaths _ _ _ p x) (z x).
Proof.
  intros. induction p. reflexivity.
Defined.

Definition transport_funapp {T} {X Y:T->Type}
           (f:∏ t, X t -> Y t) (x:∏ t, X t)
           {t t':T} (p:t = t') :
  transportf _ p ((f t)(x t))
  = (transportf (λ t, X t -> Y t) p (f t)) (transportf _ p (x t)).
Proof.
  intros. induction p. reflexivity.
Defined.

Definition helper_A {T} {Y} (P:T->Y->Type) {y y':Y} (k:∏ t, P t y) (e:y = y') t :
  transportf (λ y, P t y) e (k t)
  =
  (transportf (λ y, ∏ t, P t y) e k) t.
Proof.
  intros. induction e. reflexivity.
Defined.

Definition helper_B {T} {Y} (f:T->Y) {y y':Y} (k:∏ t, y = f t) (e:y = y') t :
  transportf (λ y, y = f t) e (k t)
  =
  (transportf (λ y, ∏ t, y = f t) e k) t.
Proof.
  intros. exact (helper_A _ k e t).
Defined.

Definition transport_invweq {T} {X Y:T->Type} (f:∏ t, (X t) ≃ (Y t))
           {t t':T} (p:t = t') :
  transportf (λ t, (Y t) ≃ (X t)) p (invweq (f t))
  =
  invweq (transportf (λ t, (X t) ≃ (Y t)) p (f t)).
Proof.
  intros. induction p. reflexivity.
Defined.

Definition transport_invmap {T} {X Y:T->Type} (f:∏ t, (X t) ≃ (Y t))
           {t t':T} (p:t=t') :
  transportf (λ t, Y t -> X t) p (invmap (f t))
  =
  invmap (transportf (λ t, (X t) ≃ (Y t)) p (f t)).
Proof.
  intros. induction p. reflexivity.
Defined.

(** *** Double transport *)

Definition transportf2 {X} {Y:X->Type} (Z:∏ x, Y x->Type)
           {x x'} (p:x = x')
           (y:Y x) (z:Z x y) : Z x' (p#y).
Proof.
  intros. induction p. exact z.
Defined.

Definition transportb2 {X} {Y:X->Type} (Z:∏ x, Y x->Type)
           {x x'} (p:x=x')
           (y':Y x') (z':Z x' y') : Z x (p#'y').
Proof.
  intros. induction p. exact z'.
Defined.

Definition maponpaths_pr1_pr2 {X} {P:X->UU} {Q:∏ x, P x->Type}
           {w w': ∑ x p, Q x p}
           (p : w = w') :
  transportf P (maponpaths pr1 p) (pr1 (pr2 w)) = pr1 (pr2 w').
Proof.
  intros. induction p. reflexivity.
Defined.

(** *** Transport a pair *)

(* (* replace this with transportf_total2 (?) : *) *)
(* Definition transportf_pair X (Y:X->Type) (Z:∏ x, Y x->Type) *)
(*            x x' (p:x = x') (y:Y x) (z:Z x y) : *)
(*   transportf (λ x, total2 (Z x)) p (tpair (Z x) y z) *)
(*   = *)
(*   tpair (Z x') (transportf Y p y) (transportf2 _ p y z). *)
(* Proof. *)
(*   intros. induction p. reflexivity. *)
(* Defined. *)

Definition transportb_pair X (Y:X->Type) (Z:∏ x, Y x->Type)
           x x' (p:x = x')
           (y':Y x') (z':Z x' y')
           (z'' : (Z x' y')) :
  transportb (λ x, total2 (Z x)) p (tpair (Z x') y' z')
  =
  tpair (Z x) (transportb Y p y') (transportb2 _ p y' z').
Proof.
  intros. induction p. reflexivity.
Defined.


Lemma transportf_total2_const (A B : UU) (P : B -> A -> UU) (b : B) (a1 a2 : A) (e : a1 = a2) (p : P b a1) :
  transportf (λ x, ∑ y, P y x) e (b,, p) = b,, transportf (P b) e p.
Proof.
  induction e.
  apply idpath.
Defined.



(** ** h-levels and paths *)

Lemma isaprop_wma_inhab X : (X -> isaprop X) -> isaprop X.
Proof.
  intros f. apply invproofirrelevance. intros x y. apply (f x).
Qed.

Lemma isaprop_wma_inhab' X : (X -> iscontr X) -> isaprop X.
Proof.
  intros f. apply isaprop_wma_inhab. intro x. apply isapropifcontr.
       apply (f x).
Qed.

Goal ∏ (X:hSet) (x y:X) (p q:x = y), p = q.
Proof.
  intros. apply setproperty.
Defined.

Goal ∏ (X:Type) (x y:X) (p q:x = y), isaset X -> p = q.
Proof.
  intros * is. apply is.
Defined.

Definition funset X (Y:hSet) : hSet
  := make_hSet (X->Y) (impredfun 2 _ _ (setproperty Y)).

Lemma eq_equalities_between_pairs (A : UU) (B : A -> UU)(x y : total2 (λ x, B x))
    (p q : x = y) (H : base_paths _ _ p = base_paths _ _ q)
    (H2 : transportf (fun p : pr1 x = pr1 y =>  transportf _ p (pr2 x) = pr2 y )
         H (fiber_paths p) = fiber_paths q) :  p = q.
Proof.
  apply (invmaponpathsweq (total2_paths_equiv _ _ _ )).
  apply (total2_paths_f (B:=(fun p : pr1 x = pr1 y =>
          transportf (λ x : A, B x) p (pr2 x) = pr2 y))
          (s  := (total2_paths_equiv B x y) p)
          (s' := (total2_paths_equiv B x y) q) H).
  assumption.
Defined.

(* perhaps consider name *)
Lemma total2_reassoc_paths {A} {B : A → UU} {C : (∑ a, B a) -> UU}
    (BC : A -> UU := λ a, ∑ b, C (a,,b))
    {a1 a2 : A} (bc1 : BC a1) (bc2 : BC a2)
    (ea : a1 = a2)
    (eb : transportf _ ea (pr1 bc1) = pr1 bc2)
    (ec : transportf C (two_arg_paths_f (*was total2_paths2*) ea eb) (pr2 bc1) = pr2 bc2)
  : transportf _ ea bc1 = bc2.
Proof.
  destruct bc1 as [b1 c1], bc2 as [b2 c2]; simpl in *.
  destruct ea. destruct eb. simpl in *. destruct ec.
  apply idpath.
Defined.

(* perhaps consider name *)
Lemma total2_reassoc_paths' {A} {B : A → UU} {C : (∑ a, B a) -> UU}
    (BC : A -> UU := λ a, ∑ b, C (a,,b))
    {a1 a2 : A} (bc1 : BC a1) (bc2 : BC a2)
    (ea : a1 = a2)
    (eb : pr1 bc1 = transportb _ ea (pr1 bc2))
    (ec : pr2 bc1 = transportb C (total2_paths2_b ea eb) (pr2 bc2))
  : bc1 = transportb BC ea bc2.
Proof.
  destruct ea, bc1 as [b1 c1], bc2 as [b2 c2].
  cbn in eb; destruct eb; cbn in ec; destruct ec.
  apply idpath.
Defined.

Section InvRot.

  (** moving pathsinv0 to the other side in equations *)

  Local Set Implicit Arguments.
  Local Unset Strict Implicit.

  Definition invrot (X:Type) (x y:X) (p:x=y) (p':y=x) : !p = p' -> p = !p'.
  Proof.
    intros e. induction e. apply pathsinv0. apply pathsinv0inv0.
  Defined.

  Definition invrot' (X:Type) (x y:X) (p:x=y) (p':y=x) : p = !p' -> !p = p'.
  Proof.
    intros e. induction (!e); clear e. apply pathsinv0inv0.
  Defined.

  Definition invrot'rot (X:Type) (x y:X) (p:x=y) (p':y=x) (e : !p = p') :
    invrot' (invrot e) = e.
  Proof.
    now induction e,p.
  Defined.

  Definition invrotrot' (X:Type) (x y:X) (p:x=y) (p':y=x) (e : p = !p') :
    invrot (invrot' e) = e.
  Proof.
    rewrite <- (pathsinv0inv0 e).
    generalize (!e); clear e.
    intros e.
    now induction e, p'.
  Defined.

End InvRot.

Section Weqpaths.

  (** a more basic approach to proving isweqpathscomp0r *)

  Local Set Implicit Arguments.
  Local Unset Strict Implicit.

  Definition hornRotation_rr {X:Type} {x y z : X} {p : x = z} {q : y = z} {r : x = y} :
    r = p @ !q ≃ r @ q = p.
  Proof.
    induction q, r; cbn. apply eqweqmap.
    apply (maponpaths (λ k, idpath x = k)). apply pathscomp0rid.
  Defined.

  Definition hornRotation_lr {X:Type} {x y z : X} {p : x = z} {q : y = z} {r : x = y} :
    q = (!r) @ p ≃ r @ q = p.
  Proof.
    induction p, r; cbn. apply idweq.
  Defined.

  Definition hornRotation_rl {X:Type} {x y z : X} {p : x = z} {q : y = z} {r : x = y} :
    p @ !q = r ≃ p = r @ q.
  Proof.
    induction q, r; cbn. apply eqweqmap.
    apply (maponpaths (λ k, k = idpath x)). apply pathscomp0rid.
  Defined.

  Definition hornRotation_ll {X:Type} {x y z : X} {p : x = z} {q : y = z} {r : x = y} :
    (!r) @ p = q ≃ p = r @ q.
  Proof.
    induction p, r; cbn. apply idweq.
  Defined.

  Corollary uniqueFiller (X:Type) (x y z : X) (p : x = z) (q : y = z) :
    ∃! r, r @ q = p.
  Proof.
    refine (@iscontrweqf (∑ r, r = p @ !q) _ _ _).
    { apply weqfibtototal; intro r. exact hornRotation_rr. }
    { apply iscontrcoconustot. }
  Defined.

  Lemma fillerEquation {X:Type} {x y z : X} {p : x = z} {q : y = z}
        (r : x = y) (k : r @ q = p) :
    @paths (∑ r, r@q=p)
           (r ,, k)
           ((p @ !q) ,, hornRotation_rr (idpath _)).
  Proof.
    apply proofirrelevancecontr. apply uniqueFiller.
  Defined.

  Corollary isweqpathscomp0r' {X : UU} (x : X) {x' x'' : X} (e' : x' = x'') :
    isweq (λ e : x = x', e @ e').
  Proof.
    (* make a direct proof of isweqpathscomp0r, without using isweq_iso *)
    intros p. use tpair. exists (p @ !e'). now apply hornRotation_rr.
    cbn. intros [q l]. apply fillerEquation.
  Defined.

  Definition transportPathTotal {X:Type} {x x':X} {Y : X -> Type} (y : Y x) (y' : Y x')
             (r : (x,,y) = (x',,y'))
             (T : ∏ (a:X) (b:Y a), Type) : T x y → T x' y'.
  Proof.
    induction (total2_paths_equiv _ _ _ r) as [p q].
    change (x=x') in p. change (transportf _ p y = y') in q.
    induction p.
    change (y=y') in q.
    induction q.
    trivial.
  Defined.

  Definition inductionOnFiller {X:Type} {x y z:X} (p:x=z) (q:y=z)
             (S := λ r:x=y, r @ q = p)
             (T : ∏ r (e : S r), Type)
             (t : T (p @ !q) (hornRotation_rr (idpath _))) :
    ∏ (r:x=y) (e : r @ q = p), T r e.
  Proof.
    intros.
    use (transportPathTotal _ t).
    apply pathsinv0.
    apply fillerEquation.
  Defined.

End Weqpaths.

Lemma transportf_paths_FlFr {A B : UU} {f g : A -> B} {x1 x2 : A}
 (p : x1 = x2) (q : f x1 = g x1)
 : transportf (λ x, f x = g x) p q = !maponpaths f p @ q @ maponpaths g p.
Proof.
 induction p. cbn.
 apply pathsinv0.
 apply pathscomp0rid.
Qed.

Lemma transportf_sec_constant
 {A B : UU} {C : A -> B -> UU}
 {x1 x2 : A} (p : x1 = x2) (f : ∏ y : B, C x1 y)
 : (transportf (λ x, ∏ y : B, C x y) p f)
   = (λ y, transportf (λ x, C x y) p (f y)).
Proof.
 induction p; reflexivity.
Qed.

(** More facts on fiber products *)
Definition path_hfp
           {X Y Z : UU}
           {f : X → Z}
           {g : Y → Z}
           {x y : hfp f g}
           (p₁ : hfpg f g x = hfpg f g y)
           (p₂ : hfpg' f g x = hfpg' f g y)
           (p₃ : commhfp f g x @ maponpaths f p₁
                 =
                 maponpaths g p₂ @ commhfp f g y)
  : x = y.
Proof.
  induction x as [ [ x₁ x₂ ] px ].
  induction y as [ [ y₁ y₂ ] py ].
  cbn in p₁, p₂.
  induction p₁, p₂.
  cbn in p₃.
  apply maponpaths.
  exact (!(pathscomp0rid _) @ p₃).
Defined.

Definition maponpaths_hfpg_path_hfp
           {X Y Z : UU}
           {f : X → Z}
           {g : Y → Z}
           {x y : hfp f g}
           (p₁ : hfpg f g x = hfpg f g y)
           (p₂ : hfpg' f g x = hfpg' f g y)
           (p₃ : commhfp f g x @ maponpaths f p₁
                 =
                 maponpaths g p₂ @ commhfp f g y)
  : maponpaths
      (hfpg f g)
      (path_hfp p₁ p₂ p₃)
    =
    p₁.
Proof.
  induction x as [ [ x₁ x₂ ] px ].
  induction y as [ [ y₁ y₂ ] py ].
  cbn in p₁, p₂.
  induction p₁, p₂.
  cbn in p₃.
  induction p₃ ; cbn.
  etrans.
  {
    apply (maponpathscomp
             (tpair (λ xx', g (pr2 xx') = f (pr1 xx')) (x₁,, x₂))
             (hfpg f g)).
  }
  apply maponpaths_for_constant_function.
Qed.

Definition maponpaths_hfpg'_path_hfp
           {X Y Z : UU}
           {f : X → Z}
           {g : Y → Z}
           {x y : hfp f g}
           (p₁ : hfpg f g x = hfpg f g y)
           (p₂ : hfpg' f g x = hfpg' f g y)
           (p₃ : commhfp f g x @ maponpaths f p₁
                 =
                 maponpaths g p₂ @ commhfp f g y)
  : maponpaths
      (hfpg' f g)
      (path_hfp p₁ p₂ p₃)
    =
    p₂.
Proof.
  induction x as [ [ x₁ x₂ ] px ].
  induction y as [ [ y₁ y₂ ] py ].
  cbn in p₁, p₂.
  induction p₁, p₂.
  cbn in p₃.
  induction p₃ ; cbn.
  etrans.
  {
    apply (maponpathscomp
             (tpair (λ xx', g (pr2 xx') = f (pr1 xx')) (x₁,, x₂))
             (hfpg' f g)).
  }
  apply maponpaths_for_constant_function.
Qed.

Definition path_hfp_eta
           {X Y Z : UU}
           {f : X → Z}
           {g : Y → Z}
           {x y : hfp f g}
           (p : x = y)
  : p
    =
    path_hfp
      (maponpaths (hfpg f g) p)
      (maponpaths (hfpg' f g) p)
      (maponpaths (λ z, _ @ z) (maponpathscomp (hfpg f g) f p)
       @ !(homotsec_natural (commhfp f g) p)
       @ maponpaths (λ z, z @ _) (!(maponpathscomp (hfpg' f g) g p))).
Proof.
  induction p.
  cbn.
  refine (!_).
  etrans.
  {
    apply maponpaths.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply pathscomp0rid.
      }
      apply pathsinv0inv0.
    }
    apply pathsinv0l.
  }
  apply idpath.
Qed.

Definition homot_hfp
           {X Y Z : UU}
           {f : X → Z} {g : Y → Z}
           {x y : hfp f g}
           {h₁ h₁' : hfpg f g x = hfpg f g y}
           (e₁ : h₁ = h₁')
           {h₂ h₂' : hfpg' f g x = hfpg' f g y}
           (e₂ : h₂ = h₂')
           (h₃ : commhfp f g x @ maponpaths f h₁ = maponpaths g h₂ @ commhfp f g y)
  : path_hfp h₁ h₂ h₃
    =
    path_hfp
      h₁' h₂'
      (maponpaths
         (λ z, _ @ maponpaths f z)
         (!e₁)
       @ h₃
       @ maponpaths
           (λ z, maponpaths g z @ _)
           e₂).
Proof.
  induction e₁ ; induction e₂.
  simpl.
  apply maponpaths.
  exact (!(pathscomp0rid _)).
Qed.

Definition homot_hfp_one_type
           {X Y Z : UU}
           (HZ : isofhlevel 3 Z)
           {f : X → Z}
           {g : Y → Z}
           {x y : hfp f g}
           (p q : x = y)
           (r₁ : maponpaths (hfpg f g) p
                 =
                 maponpaths (hfpg f g) q)
           (r₂ : maponpaths (hfpg' f g) p
                 =
                 maponpaths (hfpg' f g) q)
  : p = q.
Proof.
  refine (path_hfp_eta p @ _ @ !(path_hfp_eta q)).
  etrans.
  {
    exact (homot_hfp r₁ r₂ _).
  }
  apply maponpaths.
  apply HZ.
Qed.

Definition hfp_is_of_hlevel
           (n : nat)
           {X Y Z : UU}
           (HX : isofhlevel n X)
           (HY : isofhlevel n Y)
           (HZ : isofhlevel n Z)
           (f : X → Z)
           (g : Y → Z)
  : isofhlevel n (hfp f g).
Proof.
  use isofhleveltotal2.
  - use isofhleveldirprod.
    + exact HX.
    + exact HY.
  - simpl.
    intro x.
    apply (hlevelntosn n _ HZ).
Qed.

Definition hfp_HLevel
           (n : nat)
           {X Y Z : HLevel n}
           (f : pr1 X → pr1 Z)
           (g : pr1 Y → pr1 Z)
  : HLevel n.
Proof.
  simple refine (hfp f g ,, _).
  apply hfp_is_of_hlevel.
  - exact (pr2 X).
  - exact (pr2 Y).
  - exact (pr2 Z).
Defined.

(** Transport along a path of total2 *)
Definition transportf_total2_paths_f
           {A : UU}
           {B : A → UU}
           (C : A → UU)
           {a₁ a₂ : A}
           {b₁ : B a₁}
           {b₂ : B a₂}
           (p : a₁ = a₂)
           (q : transportf B p b₁ = b₂)
           (c₁ : C a₁)
  : transportf
      (λ z, C (pr1 z))
      (@total2_paths_f
         A B
         (a₁ ,, b₁) (a₂ ,, b₂)
         p
         q)
      c₁
    =
    transportf
      C
      p
      c₁.
Proof.
  induction p.
  induction q.
  apply idpath.
Defined.


(** Paths of products *)
Definition maponpaths_pr1_pathsdirprod
           {X Y : UU}
           {x₁ x₂ : X}
           {y₁ y₂ : Y}
           (p : x₁ = x₂)
           (q : y₁ = y₂)
  : maponpaths dirprod_pr1 (pathsdirprod p q) = p.
Proof.
  induction p, q.
  apply idpath.
Defined.

Definition maponpaths_pr2_pathsdirprod
           {X Y : UU}
           {x₁ x₂ : X}
           {y₁ y₂ : Y}
           (p : x₁ = x₂)
           (q : y₁ = y₂)
  : maponpaths dirprod_pr2 (pathsdirprod p q) = q.
Proof.
  induction p, q.
  apply idpath.
Defined.

Definition pathsdirprod_eta
           {X Y : UU}
           {x y : X × Y}
           (p : x = y)
  : p
    =
    pathsdirprod (maponpaths dirprod_pr1 p) (maponpaths dirprod_pr2 p).
Proof.
  induction p.
  apply idpath.
Defined.

Definition paths_pathsdirprod
           {X Y : UU}
           {x₁ x₂ : X}
           {y₁ y₂ : Y}
           {p₁ p₂ : x₁ = x₂}
           {q₁ q₂ : y₁ = y₂}
           (r₁ : p₁ = p₂)
           (r₂ : q₁ = q₂)
  : pathsdirprod p₁ q₁
    =
    pathsdirprod p₂ q₂.
Proof.
  induction r₁, r₂.
  apply idpath.
Defined.

(** Paths on functions *)
Definition app_fun
           {X Y : UU}
  : (X → Y) × X → Y
  := λ fx, pr1 fx (pr2 fx).

Definition app_homot
           {X Y₁ Y₂ : UU}
           {f g : Y₁ → X → Y₂}
           (p : ∏ (z : Y₁ × X), f (pr1 z) (pr2 z) = g (pr1 z) (pr2 z))
           (y : Y₁)
  : f y = g y
  := funextsec _ _ _ (λ x, p (y ,, x)).

Definition maponpaths_app_fun
           {X Y : UU}
           {fx gx : (X → Y) × X}
           (p : fx = gx)
  : maponpaths (λ (fx : (X → Y) × X), app_fun fx) p
    =
    maponpaths (λ z, z (pr2 fx)) (maponpaths dirprod_pr1 p)
    @
    maponpaths (pr1 gx) (maponpaths dirprod_pr2 p).
Proof.
  induction p.
  apply idpath.
Defined.



(** Product of a propositions with itself *)
Definition dirprod_with_prop (A : UU) (isa : isaprop A) : A × A ≃ A.
Proof.
  apply weqpr1, iscontraprop1; assumption.
Defined.

(** A variation on the above theme *)
Definition dirprod_with_prop' (A B : UU) (isa : isaprop A) : A × B × A ≃ B × A.
Proof.
  intermediate_weq ((A × B) × A).
  apply invweq, weqtotal2asstor.
  intermediate_weq (A × (A × B)).
  apply weqdirprodcomm.
  intermediate_weq ((A × A) × B).
  apply invweq, weqtotal2asstor.
  intermediate_weq (A × B).
  apply weqdirprodf.
  - apply dirprod_with_prop; assumption.
  - apply idweq.
  - apply weqdirprodcomm.
Defined.

(** ** Surjectivity *)

Section surjectivity.
  Lemma issurjective_idfun (X : UU) : issurjective (idfun X).
  Proof.
    exact(λ (x : X), hinhpr (x ,, idpath x)).
  Qed.

  Lemma issurjective_to_contr {X Y : UU}
    (x : X)
    (f : X → Y)
    (contr : iscontr Y) : issurjective f.
  Proof.
    intro; apply hinhpr.
    apply(make_hfiber f x).
    apply proofirrelevancecontr, contr.
  Qed.

  Lemma issurjective_tounit {X : UU} : X -> issurjective (@tounit X).
  Proof.
    intro x.
    apply(issurjective_to_contr x).
    apply iscontrunit.
  Qed.

  Lemma issurjective_coprodf {X Y W Z : UU}
    {f : X → W}
    {g : Y → Z}
    (fsurjective : issurjective f)
    (gsurjective : issurjective g)
    : issurjective (coprodf f g).
  Proof.
    red; apply coprod_rect.
    - intro w.
      apply(hinhfun (weqhfibercoprodf1 f g w)).
      exact(fsurjective w).
    - intro z.
      apply(hinhfun (weqhfibercoprodf2 f g z)).
      exact(gsurjective z).
  Qed.

  Lemma issurjective_dirprodf {X Y U W}
    {f : X → U}
    {g : Y → W}
    (fsurjective : issurjective f)
    (gsurjective : issurjective g)
    : issurjective (dirprodf f g).
  Proof.
    intros [u w].
    use(hinhfun2 _ (fsurjective u) (gsurjective w)).
    intros ffiber gfiber.
    use make_hfiber.
    - exact(hfiberpr1 f u ffiber
              ,, hfiberpr1 g w gfiber).
    - apply dirprodeq
      ; apply hfiberpr2.
  Qed.

  Lemma issurjective_totalfun
    {X : UU}
    (P Q : X → UU)
    (f : ∏ (x : X), P x → Q x)
    (fsurjective : ∏ (x : X), issurjective (f x))
    : issurjective (totalfun P Q f).
  Proof.
    intros [x qx].
    use(hinhfun _ (fsurjective x qx)).

    intros [px pathpx]. (* pathpx : f x px = qx, in (Q x) *)
    use make_hfiber.
    - exact(x ,, px).
    - exact(pair_path_in2 Q pathpx).
  Qed.

  Lemma issurjective_sumofmaps_1 {X A B : UU}
    (f : A -> X)
    (g : B -> X)
    (fsurjective : issurjective f)
    : issurjective (sumofmaps f g).
  Proof.
    intro x.
    use(hinhfun _ (fsurjective x)).
    intro ffiber; apply coprodofhfiberstohfiber.
    exact(inl ffiber).
  Qed.

  Lemma issurjective_sumofmaps_2 {X A B : UU}
    (f : A -> X)
    (g : B -> X)
    (gsurjective : issurjective g)
    : issurjective (sumofmaps f g).
  Proof.
    intro x.
    use(hinhfun _ (gsurjective x)).
    intro gfiber; apply coprodofhfiberstohfiber.
    exact(inr gfiber).
  Qed.
End surjectivity.
