feat(hott): remove sorry's in circle.hlean, characterize pathovers in degenerate pi's

hott/hit/circle.hlean

@@ -6,9 +6,11 @@ Authors: Floris van Doorn
 Declaration of the circle
 -/
 
-import .sphere types.bool types.eq types.int.hott types.arrow types.equiv algebra.fundamental_group algebra.hott
+import .sphere
+import types.bool types.int.hott types.equiv
+import algebra.fundamental_group algebra.hott
 
-open eq suspension bool sphere_index is_equiv equiv equiv.ops is_trunc
+open eq suspension bool sphere_index is_equiv equiv equiv.ops is_trunc pi
 
 definition circle : Type₀ := sphere 1
 
@@ -171,12 +173,8 @@ namespace circle
   begin
     induction x,
     { exact loop},
-    { exact sorry}
+    { apply concato_eq, apply pathover_eq_lr, rewrite [con.left_inv,idp_con]}
   end
-  -- circle.rec_on x loop
-  --   (calc
-  --     loop ▸ loop = loop⁻¹ ⬝ loop ⬝ loop : transport_eq_lr
-  --       ... = loop : by rewrite [con.left_inv, idp_con])
 
   definition nonidp_neq_idp : nonidp ≠ (λx, idp) :=
   assume H : nonidp = λx, idp,
@@ -202,10 +200,8 @@ namespace circle
   begin
     induction x,
     { exact power loop},
-    { apply sorry}
--- apply eq_of_homotopy, intro a,
---       refine !arrow.arrow_transport ⬝ !transport_eq_r ⬝ _,
---       rewrite [transport_code_loop_inv,power_con,succ_pred]}
+    { apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
+      rewrite [power_con,transport_code_loop]},
   end
 
   --remove this theorem after #484
@@ -222,8 +218,8 @@ namespace circle
         apply transport (λ(y : base = base), transport circle.code y _ = _),
         { exact !power_con_inv ⬝ ap (power loop) !neg_succ⁻¹},
         rewrite [▸*,@con_tr _ circle.code,transport_code_loop_inv, ↑[circle.encode] at p, p, -neg_succ]}},
-    --{ apply eq_of_homotopy, intro a, apply @is_hset.elim, esimp [circle.code,base,base1], exact _}
-      { apply sorry}
+    { apply pathover_of_tr_eq, apply eq_of_homotopy, intro a, apply @is_hset.elim,
+      esimp [circle.code,base,base1], exact _}
       --simplify after #587
   end
 

hott/init/pathover.hlean

@@ -13,7 +13,7 @@ open equiv is_equiv equiv.ops
 
 variables {A A' : Type} {B : A → Type} {C : Πa, B a → Type}
           {a a₂ a₃ a₄ : A} {p : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄}
-          {b : B a} {b₂ : B a₂} {b₃ : B a₃} {b₄ : B a₄}
+          {b b' : B a} {b₂ b₂' : B a₂} {b₃ : B a₃} {b₄ : B a₄}
           {c : C a b} {c₂ : C a₂ b₂}
 
 namespace eq
@@ -67,6 +67,12 @@ namespace eq
   definition concato (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) : b =[p ⬝ p₂] b₃ :=
   pathover.rec_on r₂ (pathover.rec_on r idpo)
 
+  definition concato_eq (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' :=
+  eq.rec_on q r
+
+  definition eq_concato (q : b = b') (r : b' =[p] b₂) : b =[p] b₂ :=
+  eq.rec_on q (λr, r) r
+
   definition inverseo (r : b =[p] b₂) : b₂ =[p⁻¹] b :=
   pathover.rec_on r idpo
 
@@ -134,6 +140,14 @@ namespace eq
     from eq.rec_on (eq_of_pathover_idp r) H,
   left_inv !pathover_idp r ▸ H2
 
+  definition rec_on_right [recursor] {P : Π⦃b₂ : B a₂⦄, b =[p] b₂ → Type}
+    {b₂ : B a₂} (r : b =[p] b₂) (H : P !pathover_tr) : P r :=
+  by cases r; exact H
+
+  definition rec_on_left [recursor] {P : Π⦃b : B a⦄, b =[p] b₂ → Type}
+    {b : B a} (r : b =[p] b₂) (H : P !tr_pathover) : P r :=
+  by cases r; exact H
+
   --pathover with fibration B' ∘ f
   definition pathover_compose (B' : A' → Type) (f : A → A') (p : a = a₂)
     (b : B' (f a)) (b₂ : B' (f a₂)) : b =[p] b₂ ≃ b =[ap f p] b₂ :=

hott/types/arrow.hlean

@@ -11,7 +11,7 @@ import types.pi
 
 open eq equiv is_equiv funext pi equiv.ops
 
-namespace arrow
+namespace pi
 
   variables {A A' : Type} {B B' : Type} {C : A → B → Type}
             {a a' a'' : A} {b b' b'' : B} {f g : A → B}
@@ -50,6 +50,34 @@ namespace arrow
     : (transport (λa, B a → C a) p f) ∼ (λb, p ▸ f (p⁻¹ ▸ b)) :=
   eq.rec_on p (λx, idp)
 
+  /- Pathovers -/
 
+  definition arrow_pathover {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
+    (r : Π(b : B a) (b' : B a') (q : b =[p] b'), f b =[p] g b') : f =[p] g :=
+  begin
+    cases p, apply pathover_idp_of_eq,
+    apply eq_of_homotopy, intro b,
+    exact eq_of_pathover_idp (r b b idpo),
+  end
 
-end arrow
+  definition arrow_pathover_left {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
+    (r : Π(b : B a), f b =[p] g (p ▸ b)) : f =[p] g :=
+  begin
+    cases p, apply pathover_idp_of_eq,
+    apply eq_of_homotopy, intro b,
+    exact eq_of_pathover_idp (r b),
+  end
+
+  definition arrow_pathover_right {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
+    (r : Π(b' : B a'), f (p⁻¹ ▸ b') =[p] g b') : f =[p] g :=
+  begin
+    cases p, apply pathover_idp_of_eq,
+    apply eq_of_homotopy, intro b,
+    exact eq_of_pathover_idp (r b),
+  end
+
+  definition arrow_pathover_constant {B : Type} {C : A → Type} {f : B → C a} {g : B → C a'}
+    {p : a = a'} (r : Π(b : B), f b =[p] g b) : f =[p] g :=
+  pi_pathover_constant r
+
+end pi

hott/types/eq.hlean

@@ -9,6 +9,8 @@ Theorems about path types (identity types)
 
 open eq sigma sigma.ops equiv is_equiv equiv.ops
 
+-- TODO: Rename transport_eq_... and pathover_eq_... to eq_transport_... and eq_pathover_...
+
 namespace eq
   /- Path spaces -/
 
@@ -99,15 +101,6 @@ namespace eq
 
   -- In the comment we give the fibration of the pathover
 
-  definition pathover_eq_r_idp (p : a1 = a2) : idp =[p] p := /-(λx, a1 = x)-/
-  by cases p; exact idpo
-
-  definition pathover_eq_l_idp (p : a1 = a2) : idp =[p] p⁻¹ := /-(λx, x = a1)-/
-  by cases p; exact idpo
-
-  definition pathover_eq_l_idp' (p : a1 = a2) : idp =[p⁻¹] p := /-(λx, x = a2)-/
-  by cases p; exact idpo
-
   definition pathover_eq_l (p : a1 = a2) (q : a1 = a3) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a3)-/
   by cases p; cases q; exact idpo
 
@@ -139,6 +132,15 @@ namespace eq
   /-(λx, x = h (f x))-/
   by cases p; rewrite [▸*,idp_con];exact idpo
 
+  definition pathover_eq_r_idp (p : a1 = a2) : idp =[p] p := /-(λx, a1 = x)-/
+  by cases p; exact idpo
+
+  definition pathover_eq_l_idp (p : a1 = a2) : idp =[p] p⁻¹ := /-(λx, x = a1)-/
+  by cases p; exact idpo
+
+  definition pathover_eq_l_idp' (p : a1 = a2) : idp =[p⁻¹] p := /-(λx, x = a2)-/
+  by cases p; exact idpo
+
   -- The Functorial action of paths is [ap].
 
   /- Equivalences between path spaces -/

hott/types/equiv.hlean

@@ -9,7 +9,7 @@ Theorems about the types equiv and is_equiv
 
 import .fiber .arrow arity .hprop_trunc
 
-open eq is_trunc sigma sigma.ops arrow pi fiber function equiv
+open eq is_trunc sigma sigma.ops pi fiber function equiv
 
 namespace is_equiv
   variables {A B : Type} (f : A → B) [H : is_equiv f]

hott/types/pi.hlean

@@ -7,7 +7,7 @@ Partially ported from Coq HoTT
 Theorems about pi-types (dependent function spaces)
 -/
 
-import types.sigma
+import types.sigma arity
 
 open eq equiv is_equiv funext sigma
 
@@ -69,15 +69,30 @@ namespace pi
     apply eq_of_pathover_idp, apply r
   end
 
-  definition pi_pathover' {C : (Σa, B a) → Type} {f : Πb, C ⟨a, b⟩} {g : Πb', C ⟨a', b'⟩}
-    {p : a = a'} (r : Π(b : B a) (b' : B a') (q : pathover B b p b'), f b =[dpair_eq_dpair p q] g b')
-    : f =[p] g :=
+  definition pi_pathover_left {f : Πb, C a b} {g : Πb', C a' b'} {p : a = a'}
+    (r : Π(b : B a), f b =[apo011 C p !pathover_tr] g (p ▸ b)) : f =[p] g :=
   begin
     cases p, apply pathover_idp_of_eq,
     apply eq_of_homotopy, intro b,
-    apply (@eq_of_pathover_idp _ C), exact (r b b (pathover.idpatho b)),
+    apply eq_of_pathover_idp, apply r
   end
 
+  definition pi_pathover_right {f : Πb, C a b} {g : Πb', C a' b'} {p : a = a'}
+    (r : Π(b' : B a'), f (p⁻¹ ▸ b') =[apo011 C p !tr_pathover] g b') : f =[p] g :=
+  begin
+    cases p, apply pathover_idp_of_eq,
+    apply eq_of_homotopy, intro b,
+    apply eq_of_pathover_idp, apply r
+  end
+
+  definition pi_pathover_constant {C : A → A' → Type} {f : Π(b : A'), C a b}
+    {g : Π(b : A'), C a' b} {p : a = a'}
+    (r : Π(b : A'), f b =[p] g b) : f =[p] g :=
+  begin
+    cases p, apply pathover_idp_of_eq,
+    apply eq_of_homotopy, intro b,
+    exact eq_of_pathover_idp (r b),
+  end
 
   /- Maps on paths -/
 

hott/types/squareover.hlean

@@ -6,7 +6,7 @@ Author: Floris van Doorn
 Theorems about squareovers
 -/
 
-import cubical.pathover cubical.square
+import types.square
 
 open eq equiv is_equiv equiv.ops
 
@@ -41,6 +41,4 @@ namespace cubical
   definition idsquareo [reducible] [constructor] (b₀₀ : B a₀₀) := @squareover.idsquareo A a₀₀ B b₀₀
   definition idso [reducible] [constructor] := @squareover.idsquareo A a₀₀ B b₀₀
 
-
-
 end cubical