Merge pull request #2253 from Alizter/more-transport_paths

reorganise transport_paths lemmas and add tests

Author: Alizter

test/Tactics/transport_paths.v

@@ -0,0 +1,202 @@
+From HoTT Require Import Basics.Overture Basics.Tactics Types.Paths.
+
+
+(** Here we test the [transport_paths FlFr] style tactics. *)
+
+(** *** 0 functions *)
+
+(** Transporting the left *)
+Definition transport_paths_l {A : Type} {x1 x2 y : A} (p : x1 = x2) (q : x1 = y)
+  (r : x2 = y)
+  (expected : p^ @ q = r)
+  : transport (fun x => x = y) p q = r.
+Proof.
+  by transport_paths l.
+Defined.
+
+(** Transporting both *)
+Definition transport_paths_lr {A : Type} {x1 x2 : A} (p : x1 = x2) (q : x1 = x1)
+  (r : x2 = x2)
+  (expected : p @ r = q @ p)
+  : transport (fun x => x = x) p q = r.
+Proof.
+  by transport_paths lr.
+Defined.
+
+(** Transporting the right *)
+Definition transport_paths_r {A : Type} {x y1 y2 : A} (p : y1 = y2) (q : x = y1)
+  (r : x = y2)
+  (expected : q @ p = r)
+  : transport (fun y => x = y) p q = r.
+Proof.
+  by transport_paths r.
+Defined.
+
+(** *** 1 function *)
+
+Definition transport_paths_Fl {A B : Type} {f : A -> B} {x1 x2 : A} {y : B}
+  (p : x1 = x2) (q : f x1 = y)
+  (r : f x2 = y)
+  (expected : q = ap f p @ r)
+  : transport (fun x => f x = y) p q = r.
+Proof.
+  by transport_paths Fl.
+Defined.
+
+Definition transport_paths_Flr {A : Type} {f : A -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : f x1 = x1)
+  (r : f x2 = x2)
+  (expected : ap f p @ r = q @ p) 
+  : transport (fun x => f x = x) p q = r.
+Proof.
+  by transport_paths Flr.
+Defined.
+
+Definition transport_paths_lFr {A : Type} {f : A -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : x1 = f x1)
+  (r : x2 = f x2)
+  (expected : p @ r = q @ ap f p)
+  : transport (fun x => x = f x) p q = r.
+Proof.
+  by transport_paths lFr.
+Defined.
+
+Definition transport_paths_Fr {A B : Type} {g : A -> B} {y1 y2 : A} {x : B}
+  (p : y1 = y2) (q : x = g y1)
+  (r : x = g y2)
+  (expected : q @ ap g p = r)
+  : transport (fun y => x = g y) p q = r.
+Proof.
+  by transport_paths Fr.
+Defined.
+
+(** *** 2 functions *)
+
+Definition transport_paths_FFl {A B C : Type} {f : A -> B} {g : B -> C}
+  {x1 x2 : A} {y : C} (p : x1 = x2) (q : g (f x1) = y)
+  (r : g (f x2) = y)
+  (expected : q = ap g (ap f p) @ r)
+  : transport (fun x => g (f x) = y) p q = r.
+Proof.
+  by transport_paths FFl.
+Defined.
+
+Definition transport_paths_FFlr {A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : g (f x1) = x1)
+  (r : g (f x2) = x2)
+  (expected : ap g (ap f p) @ r = q @ p)
+  : transport (fun x => g (f x) = x) p q = r.
+Proof.
+  by transport_paths FFlr.
+Defined.
+
+Definition transport_paths_FlFr {A B : Type} {f g : A -> B} {x1 x2 : A}
+  (p : x1 = x2) (q : f x1 = g x1)
+  (r : f x2 = g x2)
+  (expected : ap f p @ r = q @ ap g p)
+  : transport (fun x => f x = g x) p q = r.
+Proof.
+  by transport_paths FlFr.
+Defined.
+
+Definition transport_paths_lFFr {A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : x1 = g (f x1))
+  (r : x2 = g (f x2))
+  (expected : p @ r = q @ ap g (ap f p))
+  : transport (fun x => x = g (f x)) p q = r.
+Proof.
+  by transport_paths lFFr.
+Defined.
+
+Definition transport_paths_FFr {A B C : Type} {f : A -> B} {g : B -> C}
+  {x1 x2 : A} {y : C} (p : x1 = x2) (q : y = g (f x1))
+  (r : y = g (f x2))
+  (expected : q @ ap g (ap f p) = r)
+  : transport (fun x => y = g (f x)) p q = r.
+Proof.
+  by transport_paths FFr.
+Defined.
+
+(** *** 3 functions *)
+
+Definition transport_paths_FFFl {A B C D : Type}
+  {f : A -> B} {g : B -> C} {h : C -> D} {x1 x2 : A} {y : D}
+  (p : x1 = x2) (q : h (g (f x1)) = y)
+  (r : h (g (f x2)) = y)
+  (expected : q = ap h (ap g (ap f p)) @ r)
+  : transport (fun x => h (g (f x)) = y) p q = r.
+Proof.
+  by transport_paths FFFl.
+Defined.
+
+Definition transport_paths_FFFlr {A B C : Type}
+  {f : A -> B} {g : B -> C} {h : C -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : h (g (f x1)) = x1)
+  (r : h (g (f x2)) = x2)
+  (expected : ap h (ap g (ap f p)) @ r = q @ p)
+  : transport (fun x => h (g (f x)) = x) p q = r.
+Proof.
+  by transport_paths FFFlr.
+Defined.
+
+Definition transport_paths_FFlFr {A B C : Type}
+  {f : A -> B} {g : B -> C} {h : A -> C} {x1 x2 : A}
+  (p : x1 = x2) (q : g (f x1) = h x1)
+  (r : g (f x2) = h x2)
+  (expected : ap g (ap f p) @ r = q @ ap h p)
+  : transport (fun x => g (f x) = h x) p q = r.
+Proof.
+  by transport_paths FFlFr.
+Defined.
+
+Definition transport_paths_FlFFr {A B C : Type}
+  {f : A -> C} {g : B -> C} {h : A -> B} {x1 x2 : A}
+  (p : x1 = x2) (q : f x1 = g (h x1))
+  (r : f x2 = g (h x2))
+  (expected : ap f p @ r = q @ ap g (ap h p))
+  : transport (fun x => f x = g (h x)) p q = r.
+Proof.
+  by transport_paths FlFFr.
+Defined.
+
+Definition transport_paths_lFFFr {A B C : Type}
+  {f : A -> B} {g : B -> C} {h : C -> A} {x1 x2 : A}
+  (p : x1 = x2) (q : x1 = h (g (f x1)))
+  (r : x2 = h (g (f x2)))
+  (expected : p @ r = q @ ap h (ap g (ap f p)))
+  : transport (fun x => x = h (g (f x))) p q = r.
+Proof.
+  by transport_paths lFFFr.
+Defined.
+
+Definition transport_paths_FFFr {A B C D : Type}
+  {f : A -> B} {g : B -> C} {h : C -> D} {x1 x2 : A} {y : D}
+  (p : x1 = x2) (q : y = h (g (f x1)))
+  (r : y = h (g (f x2)))
+  (expected : q @ ap h (ap g (ap f p)) = r)
+  : transport (fun x => y = h (g (f x))) p q = r.
+Proof.
+  by transport_paths FFFr.
+Defined.
+
+(** *** 4 functions *)
+
+Definition transport_paths_FFFlFr {A B C D : Type}
+  {f : A -> B} {g : B -> C} {h : C -> D} {k : A -> D} {x1 x2 : A}
+  (p : x1 = x2) (q : h (g (f x1)) = k x1)
+  (r : h (g (f x2)) = k x2)
+  (expected : ap h (ap g (ap f p)) @ r = q @ ap k p)
+  : transport (fun x => h (g (f x)) = k x) p q = r.
+Proof.
+  by transport_paths FFFlFr.
+Defined.
+
+Definition transport_paths_FFlFFr {A B B' C : Type}
+  {f : A -> B} {f' : A -> B'} {g : B -> C} {g' : B' -> C} {x1 x2 : A}
+  (p : x1 = x2) (q : g (f x1) = g' (f' x1))
+  (r : g (f x2) = g' (f' x2))
+  (expected : ap g (ap f p) @ r = q @ ap g' (ap f' p))
+  : transport (fun x => g (f x) = g' (f' x)) p q = r.
+Proof.
+  by transport_paths FFlFFr.
+Defined.