feat(library/standard/data/quotient): import quotient from lean 0.1

library/standard/data/prod.lean

@@ -1,6 +1,6 @@
 -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 -- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Leonardo de Moura
+-- Author: Leonardo de Moura, Jeremy Avigad
 
 import logic.classes.inhabited logic.connectives.eq
 
@@ -32,11 +32,11 @@ section
   theorem prod_ext (p : prod A B) : pair (pr1 p) (pr2 p) = p :=
   pair_destruct p (λx y, refl (x, y))
 
-  theorem pair_eq {p1 p2 : prod A B} (H1 : pr1 p1 = pr1 p2) (H2 : pr2 p1 = pr2 p2) : p1 = p2 :=
+  theorem pair_eq {a1 a2 : A} {b1 b2 : B} (H1 : a1 = a2) (H2 : b1 = b2) : (a1, b1) = (a2, b2) :=
-  calc p1  = pair (pr1 p1) (pr2 p1) : symm (prod_ext p1)
+  subst H1 (subst H2 (refl _))
-       ... = pair (pr1 p2) (pr2 p1) : {H1}
+
-       ... = pair (pr1 p2) (pr2 p2) : {H2}
+  theorem prod_eq {p1 p2 : prod A B} : ∀ (H1 : pr1 p1 = pr1 p2) (H2 : pr2 p1 = pr2 p2), p1 = p2 :=
-       ... = p2                        : prod_ext p2
+  pair_destruct p1 (take a1 b1, pair_destruct p2 (take a2 b2 H1 H2, pair_eq H1 H2))
 
   theorem prod_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (prod A B) :=
   inhabited_elim H1 (λa, inhabited_elim H2 (λb, inhabited_intro (pair a b)))

library/standard/data/quotient.lean

@@ -0,0 +1,599 @@
+-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Floris van Doorn
+
+import logic tools.tactic .subtype logic.connectives.cast struc.relation data.prod
+import logic.connectives.instances
+
+-- for now: to use substitution (iff_to_eq)
+import logic.axioms.classical
+
+-- for the last section
+import logic.axioms.hilbert logic.axioms.funext
+
+using relation prod tactic eq_proofs
+
+
+-- temporary: substiution for iff
+theorem substi {a b : Prop} {P : Prop → Prop} (H1 : a ↔ b) (H2 : P a) : P b :=
+subst (iff_to_eq H1) H2
+
+theorem transi {a b c : Prop} (H1 : a ↔ b) (H2 : b ↔ c) : a ↔ c :=
+eq_to_iff (trans (iff_to_eq H1) (iff_to_eq H2))
+
+theorem symmi {a b : Prop} (H : a ↔ b) : b ↔ a :=
+eq_to_iff (symm (iff_to_eq H))
+
+-- until we have the simplifier...
+definition simp : tactic := apply @sorry
+
+
+-- TODO: find a better name, and move to logic.connectives.basic
+theorem and_inhabited_left {a : Prop} (b : Prop) (Ha : a) : a ∧ b ↔ b :=
+iff_intro (take Hab, and_elim_right Hab) (take Hb, and_intro Ha Hb)
+
+
+-- auxliary facts about products
+-- -----------------------------
+
+-- TODO: move to data.prod?
+
+-- ### flip
+
+definition flip {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)
+
+theorem flip_def {A B : Type} (a : A × B) : flip a = pair (pr2 a) (pr1 a) := refl (flip a)
+
+theorem flip_pair {A B : Type} (a : A) (b : B) : flip (pair a b) = pair b a := rfl
+
+theorem flip_pr1 {A B : Type} (a : A × B) : pr1 (flip a) = pr2 a := rfl
+
+theorem flip_pr2 {A B : Type} (a : A × B) : pr2 (flip a) = pr1 a := rfl
+
+theorem flip_flip {A B : Type} (a : A × B) : flip (flip a) = a := 
+pair_destruct a (take x y, rfl)
+
+theorem P_flip {A B : Type} {P : A → B → Prop} {a : A × B} (H : P (pr1 a) (pr2 a))
+  : P (pr2 (flip a)) (pr1 (flip a)) :=
+(symm (flip_pr1 a)) ▸ (symm (flip_pr2 a)) ▸ H
+
+theorem flip_inj {A B : Type} {a b : A × B} (H : flip a = flip b) : a = b :=
+have H2 : flip (flip a) = flip (flip b), from congr2 flip H,
+show a = b, from (flip_flip a) ▸ (flip_flip b) ▸ H2
+
+-- ### coordinatewise unary maps
+
+definition map_pair {A B : Type} (f : A → B) (a : A × A) : B × B := 
+pair (f (pr1 a)) (f (pr2 a))
+
+theorem map_pair_def {A B : Type} (f : A → B) (a : A × A)
+  : map_pair f a = pair (f (pr1 a)) (f (pr2 a)) := 
+rfl
+
+theorem map_pair_pair {A B : Type} (f : A → B) (a a' : A)
+  : map_pair f (pair a a') = pair (f a) (f a') := 
+(pr1_pair a a') ▸ (pr2_pair a a') ▸ (rfl)
+
+theorem map_pair_pr1 {A B : Type} (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a)
+:= pr1_pair _ _
+
+theorem map_pair_pr2 {A B : Type} (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a)
+:= pr2_pair _ _
+
+-- ### coordinatewise binary maps
+
+definition map_pair2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : C × C
+:= pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b))
+
+theorem map_pair2_def {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B)
+  : map_pair2 f a b = pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b)) := rfl
+
+theorem map_pair2_pair {A B C : Type} (f : A → B → C) (a a' : A) (b b' : B)
+  : map_pair2 f (pair a a') (pair b b') = pair (f a b) (f a' b') :=
+calc
+  map_pair2 f (pair a a') (pair b b')
+	= pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) (pr2 (pair b b')))
+	    : {pr1_pair b b'}
+    ... = pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) b') : {pr2_pair b b'}
+    ... = pair (f (pr1 (pair a a')) b) (f a' b') : {pr2_pair a a'}
+    ... = pair (f a b) (f a' b') : {pr1_pair a a'}
+
+theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B)
+  : pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := pr1_pair _ _
+
+theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B)
+  : pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := pr2_pair _ _
+
+theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B)
+  : flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) :=
+  have Hx : pr1 (flip (map_pair2 f a b)) =  pr1 (map_pair2 f (flip a) (flip b)), from
+    calc
+      pr1 (flip (map_pair2 f a b)) = pr2 (map_pair2 f a b) : flip_pr1 _
+        ... = f (pr2 a) (pr2 b) : map_pair2_pr2 f a b
+        ... = f (pr1 (flip a)) (pr2 b) : {symm (flip_pr1 a)}
+        ... = f (pr1 (flip a)) (pr1 (flip b)) : {symm (flip_pr1 b)}
+        ... = pr1 (map_pair2 f (flip a) (flip b)) : symm (map_pair2_pr1 f _ _),
+  have Hy : pr2 (flip (map_pair2 f a b)) =  pr2 (map_pair2 f (flip a) (flip b)), from
+    calc
+      pr2 (flip (map_pair2 f a b)) = pr1 (map_pair2 f a b) : flip_pr2 _
+        ... = f (pr1 a) (pr1 b) : map_pair2_pr1 f a b
+        ... = f (pr2 (flip a)) (pr1 b) : {symm (flip_pr2 a)}
+        ... = f (pr2 (flip a)) (pr2 (flip b)) : {symm (flip_pr2 b)}
+        ... = pr2 (map_pair2 f (flip a) (flip b)) : symm (map_pair2_pr2 f _ _),
+  pair_eq Hx Hy
+
+-- add_rewrite flip_pr1 flip_pr2 flip_pair
+-- add_rewrite map_pair_pr1 map_pair_pr2 map_pair_pair
+-- add_rewrite map_pair2_pr1 map_pair2_pr2 map_pair2_pair
+
+theorem map_pair2_comm {A B : Type} {f : A → A → B} (Hcomm : ∀a b : A, f a b = f b a)
+  (v w : A × A) : map_pair2 f v w = map_pair2 f w v :=
+have Hx : pr1 (map_pair2 f v w) = pr1 (map_pair2 f w v), from
+  calc
+    pr1 (map_pair2 f v w) = f (pr1 v) (pr1 w) : map_pair2_pr1 f v w
+      ... = f (pr1 w) (pr1 v) : Hcomm _ _
+      ... = pr1 (map_pair2 f w v) : symm (map_pair2_pr1 f w v),
+have Hy : pr2 (map_pair2 f v w) = pr2 (map_pair2 f w v), from
+  calc
+    pr2 (map_pair2 f v w) = f (pr2 v) (pr2 w) : map_pair2_pr2 f v w
+      ... = f (pr2 w) (pr2 v) : Hcomm _ _
+      ... = pr2 (map_pair2 f w v) : symm (map_pair2_pr2 f w v),
+pair_eq Hx Hy
+
+theorem map_pair2_assoc {A : Type} {f : A → A → A}
+    (Hassoc : ∀a b c : A, f (f a b) c = f a (f b c)) (u v w : A × A) : 
+  map_pair2 f (map_pair2 f u v) w = map_pair2 f u (map_pair2 f v w) :=
+have Hx : pr1 (map_pair2 f (map_pair2 f u v) w) =
+	  pr1 (map_pair2 f u (map_pair2 f v w)), from
+  calc
+     pr1 (map_pair2 f (map_pair2 f u v) w)
+	  = f (pr1 (map_pair2 f u v)) (pr1 w) : map_pair2_pr1 f _ _
+      ... = f (f (pr1 u) (pr1 v)) (pr1 w) : {map_pair2_pr1 f _ _}
+      ... = f (pr1 u) (f (pr1 v) (pr1 w)) : Hassoc (pr1 u) (pr1 v) (pr1 w)
+      ... = f (pr1 u) (pr1 (map_pair2 f v w)) : {symm (map_pair2_pr1 f _ _)}
+      ... = pr1 (map_pair2 f u (map_pair2 f v w)) : symm (map_pair2_pr1 f _ _),
+have Hy : pr2 (map_pair2 f (map_pair2 f u v) w) =
+	  pr2 (map_pair2 f u (map_pair2 f v w)), from
+  calc
+     pr2 (map_pair2 f (map_pair2 f u v) w)
+	  = f (pr2 (map_pair2 f u v)) (pr2 w) : map_pair2_pr2 f _ _
+      ... = f (f (pr2 u) (pr2 v)) (pr2 w) : {map_pair2_pr2 f _ _}
+      ... = f (pr2 u) (f (pr2 v) (pr2 w)) : Hassoc (pr2 u) (pr2 v) (pr2 w)
+      ... = f (pr2 u) (pr2 (map_pair2 f v w)) : {symm (map_pair2_pr2 f _ _)}
+      ... = pr2 (map_pair2 f u (map_pair2 f v w)) : symm (map_pair2_pr2 f _ _),
+pair_eq Hx Hy
+
+theorem map_pair2_id_right {A B : Type} {f : A → B → A} {e : B} (Hid : ∀a : A, f a e = a) 
+  (v : A × A) : map_pair2 f v (pair e e) = v :=
+have Hx : pr1 (map_pair2 f v (pair e e)) = pr1 v, from
+  (calc
+    pr1 (map_pair2 f v (pair e e)) = f (pr1 v) (pr1 (pair e e)) : by simp
+      ... = f (pr1 v) e : by simp
+      ... = pr1 v : Hid (pr1 v)),
+have Hy : pr2 (map_pair2 f v (pair e e)) = pr2 v, from
+  (calc
+    pr2 (map_pair2 f v (pair e e)) = f (pr2 v) (pr2 (pair e e)) : by simp
+      ... = f (pr2 v) e : by simp
+      ... = pr2 v : Hid (pr2 v)),
+prod_eq Hx Hy
+
+theorem map_pair2_id_left {A B : Type} {f : B → A → A} {e : B} (Hid : ∀a : A, f e a = a)
+  (v : A × A) : map_pair2 f (pair e e) v = v :=
+have Hx : pr1 (map_pair2 f (pair e e) v) = pr1 v, from
+   calc
+    pr1 (map_pair2 f (pair e e) v) = f (pr1 (pair e e)) (pr1 v) : by simp
+      ... = f e (pr1 v) : by simp
+      ... = pr1 v : Hid (pr1 v),
+have Hy : pr2 (map_pair2 f (pair e e) v) = pr2 v, from
+   calc
+    pr2 (map_pair2 f (pair e e) v) = f (pr2 (pair e e)) (pr2 v) : by simp
+      ... = f e (pr2 v) : by simp
+      ... = pr2 v : Hid (pr2 v),
+prod_eq Hx Hy
+
+opaque_hint (hiding flip map_pair map_pair2)
+
+
+-- Theory data.quotient
+-- ====================
+
+namespace quotient
+
+using subtype
+
+-- definition and basics
+-- ---------------------
+
+-- TODO: make this a structure
+definition is_quotient {A B : Type} (R : A → A → Prop) (abs : A → B) (rep : B → A) : Prop :=
+  (∀b, abs (rep b) = b) ∧
+  (∀b, R (rep b) (rep b)) ∧
+  (∀r s, R r s ↔ (R r r ∧ R s s ∧ abs r = abs s))
+
+theorem intro {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (H1 : ∀b, abs (rep b) = b) (H2 : ∀b, R (rep b) (rep b))
+  (H3 : ∀r s, R r s ↔ (R r r ∧ R s s ∧ abs r = abs s)) : is_quotient R abs rep := 
+and_intro H1 (and_intro H2 H3)
+
+-- theorem intro_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+--   (H1 : reflexive R) (H2 : ∀b, abs (rep b) = b)
+--   (H3 : ∀r s, R r s ↔ abs r = abs s) : is_quotient R abs rep :=
+-- intro
+--   H2
+--   (take b, H1 (rep b))
+--   (take r s,
+--     have H4 : R r s ↔ R s s ∧ abs r = abs s,
+--       from 
+--         gensubst.subst (relation.operations.symm (and_inhabited_left _ (H1 s))) (H3 r s),
+--     gensubst.subst (relation.operations.symm (and_inhabited_left _ (H1 r))) H4)
+
+-- these work now, but the above still does not
+-- theorem test (a b c : Prop) (P : Prop → Prop) (H1 : a ↔ b) (H2 : c ∧ a) : c ∧ b :=
+-- gensubst.subst H1 H2
+
+-- theorem test2 {A : Type} {R : A → A → Prop} (Q : Prop) (r s : A) 
+--   (H3 : R r s ↔ Q) (H1 : R s s) : Q ↔ (R s s ∧ Q) :=
+-- relation.operations.symm (and_inhabited_left Q H1)
+
+-- theorem test3 {A : Type} {R : A → A → Prop} (Q : Prop) (r s : A) 
+--   (H3 : R r s ↔ Q) (H1 : R s s) : R r s ↔ (R s s ∧ Q) :=
+-- gensubst.subst (test2 Q r s H3 H1) H3
+
+-- theorem test4 {A : Type} {R : A → A → Prop} (Q : Prop) (r s : A) 
+--   (H3 : R r s ↔ Q) (H1 : R s s) : R r s ↔ (R s s ∧ Q) :=
+-- gensubst.subst (relation.operations.symm (and_inhabited_left Q H1)) H3
+
+theorem intro_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (H1 : reflexive R) (H2 : ∀b, abs (rep b) = b)
+  (H3 : ∀r s, R r s ↔ abs r = abs s) : is_quotient R abs rep :=
+intro
+  H2
+  (take b, H1 (rep b))
+  (take r s,
+    have H4 : R r s ↔ R s s ∧ abs r = abs s,
+      from 
+        substi (symmi (and_inhabited_left _ (H1 s))) (H3 r s),
+    substi (symmi (and_inhabited_left _ (H1 r))) H4)
+
+theorem abs_rep {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (b : B) :  abs (rep b) = b := 
+and_elim_left Q b
+
+theorem refl_rep {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (b : B) : R (rep b) (rep b) := 
+and_elim_left (and_elim_right Q) b
+
+theorem R_iff {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (r s : A) : R r s ↔ (R r r ∧ R s s ∧ abs r = abs s) :=
+and_elim_right (and_elim_right Q) r s
+
+theorem refl_left {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {r s : A} (H : R r s) : R r r :=
+and_elim_left (iff_elim_left (R_iff Q r s) H)
+
+theorem refl_right {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {r s : A} (H : R r s) : R s s :=
+and_elim_left (and_elim_right (iff_elim_left (R_iff Q r s) H))
+
+theorem eq_abs {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {r s : A} (H : R r s) : abs r = abs s :=
+and_elim_right (and_elim_right (iff_elim_left (R_iff Q r s) H))
+
+theorem R_intro {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {r s : A} (H1 : R r r) (H2 : R s s) (H3 : abs r = abs s) : R r s :=
+iff_elim_right (R_iff Q r s) (and_intro H1 (and_intro H2 H3))
+
+theorem R_intro_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (H1 : reflexive R) {r s : A} (H2 : abs r = abs s) : R r s :=
+iff_elim_right (R_iff Q r s) (and_intro (H1 r) (and_intro (H1 s) H2))
+
+theorem rep_eq {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {a b : B} (H : R (rep a) (rep b)) : a = b :=
+calc
+  a = abs (rep a) : symm (abs_rep Q a)
+    ... = abs (rep b) : {eq_abs Q H}
+    ... = b : abs_rep Q b
+
+theorem R_rep_abs {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {a : A} (H : R a a) : R a (rep (abs a)) :=
+have H3 : abs a = abs (rep (abs a)), from symm (abs_rep Q (abs a)),
+R_intro Q H (refl_rep Q (abs a)) H3
+
+theorem quotient_imp_symm {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) : symmetric R :=
+take a b : A,
+assume H : R a b,
+have Ha : R a a, from refl_left Q H,
+have Hb : R b b, from refl_right Q H,
+have Hab : abs b = abs a, from symm (eq_abs Q H),
+R_intro Q Hb Ha Hab
+
+theorem quotient_imp_trans {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) : transitive R :=
+take a b c : A,
+assume Hab : R a b,
+assume Hbc : R b c,
+have Ha : R a a, from refl_left Q Hab,
+have Hc : R c c, from refl_right Q Hbc,
+have Hac : abs a = abs c, from trans (eq_abs Q Hab) (eq_abs Q Hbc),
+R_intro Q Ha Hc Hac
+
+-- recursion
+-- ---------
+
+-- (maybe some are superfluous)
+
+definition rec {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : B → Type} (f : forall (a : A), C (abs a)) (b : B) : C b :=
+eq_rec_on (abs_rep Q b) (f (rep b))
+
+theorem comp {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : B → Type} {f : forall (a : A), C (abs a)}
+  (H : forall (r s : A) (H' : R r s), eq_rec_on (eq_abs Q H') (f r) = f s)
+  {a : A} (Ha : R a a) : rec Q f (abs a) = f a :=
+have H2 [fact] : R a (rep (abs a)), from R_rep_abs Q Ha,
+calc
+  rec Q f (abs a) =  eq_rec_on _ (f (rep (abs a))) : rfl 
+    ... = eq_rec_on _ (eq_rec_on _ (f a)) : {symm (H _ _ H2)}
+    ... = eq_rec_on _ (f a) : eq_rec_on_compose _ _ _
+    ... = f a : eq_rec_on_id _ _
+
+definition rec_constant {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : Type} (f : A → C) (b : B) : C :=
+f (rep b)
+
+theorem comp_constant {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : Type} {f : A → C}
+  (H : forall (r s : A) (H' : R r s), f r = f s)
+  {a : A} (Ha : R a a) : rec_constant Q f (abs a) = f a :=
+have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
+calc
+  rec_constant Q f (abs a) = f (rep (abs a)) : rfl
+    ... = f a : {symm (H _ _ H2)}
+
+definition rec_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : Type} (f : A → A → C) (b c : B) : C :=
+f (rep b) (rep c)
+
+theorem comp_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {C : Type} {f : A → A → C}
+  (H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), f a b = f a' b')
+  {a b : A} (Ha : R a a) (Hb : R b b) : rec_binary Q f (abs a) (abs b) = f a b :=
+have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
+have H3 : R b (rep (abs b)), from R_rep_abs Q Hb,
+calc
+  rec_binary Q f (abs a) (abs b) = f (rep (abs a))  (rep (abs b)) : rfl
+    ... = f a b : {symm (H _ _ _ _ H2 H3)}
+
+theorem comp_binary_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (Hrefl : reflexive R) {C : Type} {f : A → A → C}
+  (H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), f a b = f a' b')
+  (a b : A) : rec_binary Q f (abs a) (abs b) = f a b :=
+comp_binary Q H (Hrefl a) (Hrefl b)
+
+definition quotient_map {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (f : A → A) (b : B) : B :=
+abs (f (rep b))
+
+theorem comp_quotient_map {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {f : A → A}
+  (H : forall (a a' : A) (Ha : R a a'), R (f a) (f a'))
+  {a : A} (Ha : R a a) : quotient_map Q f (abs a) = abs (f a) :=
+have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
+have H3 : R (f a) (f (rep (abs a))), from H _ _ H2,
+have H4 : abs (f a) = abs (f (rep (abs a))), from eq_abs Q H3,
+symm H4
+
+definition quotient_map_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) (f : A → A → A) (b c : B) : B :=
+abs (f (rep b) (rep c))
+
+theorem comp_quotient_map_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
+  (Q : is_quotient R abs rep) {f : A → A → A}
+  (H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), R (f a b) (f a' b'))
+  {a b : A} (Ha : R a a) (Hb : R b b) : quotient_map_binary Q f (abs a) (abs b) = abs (f a b) :=
+have Ha2 : R a (rep (abs a)), from R_rep_abs Q Ha,
+have Hb2 : R b (rep (abs b)), from R_rep_abs Q Hb,
+have H2 : R (f a b) (f (rep (abs a)) (rep (abs b))), from H _ _ _ _ Ha2 Hb2,
+symm (eq_abs Q H2)
+
+theorem comp_quotient_map_binary_refl {A B : Type} {R : A → A → Prop} (Hrefl : reflexive R)
+  {abs : A → B} {rep : B → A} (Q : is_quotient R abs rep) {f : A → A → A}
+  (H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), R (f a b) (f a' b'))
+  (a b : A) : quotient_map_binary Q f (abs a) (abs b) = abs (f a b) :=
+comp_quotient_map_binary Q H (Hrefl a) (Hrefl b)
+
+opaque_hint (hiding rec rec_constant rec_binary quotient_map quotient_map_binary)
+
+
+-- image
+-- -----
+
+-- has to be an abbreviation, so that fun_image_definition below will typecheck outside
+-- the file
+abbreviation image {A B : Type} (f : A → B) := subtype (fun b, ∃a, f a = b)
+
+theorem image_inhabited {A B : Type} (f : A → B) (H : inhabited A) : inhabited (image f) :=
+inhabited_intro (tag (f (default A)) (exists_intro (default A) rfl))
+
+theorem image_inhabited2 {A B : Type} (f : A → B) (a : A) : inhabited (image f) :=
+image_inhabited f (inhabited_intro a)
+
+definition fun_image {A B : Type} (f : A → B) (a : A) : image f :=
+tag (f a) (exists_intro a rfl)
+
+theorem fun_image_def {A B : Type} (f : A → B) (a : A) : 
+  fun_image f a = tag (f a) (exists_intro a rfl) := rfl
+
+theorem elt_of_fun_image {A B : Type} (f : A → B) (a : A) : elt_of (fun_image f a) = f a :=
+elt_of_tag _ _
+
+theorem image_elt_of {A B : Type} {f : A → B} (u : image f) : ∃a, f a = elt_of u :=
+has_property u
+
+theorem fun_image_surj {A B : Type} {f : A → B} (u : image f) : ∃a, fun_image f a = u :=
+subtype_destruct u 
+  (take (b : B) (H : ∃a, f a = b),
+    obtain a (H': f a = b), from H,
+    (exists_intro a (tag_eq H')))
+
+theorem image_tag {A B : Type} {f : A → B} (u : image f) : ∃a H, tag (f a) H = u :=
+obtain a (H : fun_image f a = u), from fun_image_surj u,
+exists_intro a (exists_intro (exists_intro a rfl) H)
+
+theorem fun_image_eq {A B : Type} (f : A → B) (a a' : A)
+  : (f a = f a') ↔ (fun_image f a = fun_image f a') :=
+iff_intro
+  (assume H : f a = f a', tag_eq H)
+  (assume H : fun_image f a = fun_image f a',
+    subst (subst (congr2 elt_of H) (elt_of_fun_image f a)) (elt_of_fun_image f a'))
+
+theorem idempotent_image_elt_of {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f)
+  : fun_image f (elt_of u) = u :=
+obtain (a : A) (Ha : fun_image f a = u), from fun_image_surj u,
+calc
+  fun_image f (elt_of u) = fun_image f (elt_of (fun_image f a)) : {symm Ha}
+    ... = fun_image f (f a) : {elt_of_fun_image f a}
+    ... = fun_image f a : {iff_elim_left (fun_image_eq f (f a) a) (H a)}
+    ... = u : Ha
+
+theorem idempotent_image_fix {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f)
+  : f (elt_of u) = elt_of u :=
+obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
+calc
+  f (elt_of u) = f (f a) : {symm Ha}
+    ... = f a : H a
+    ... = elt_of u : Ha
+
+
+-- construct quotient from representative map
+-- ------------------------------------------
+
+theorem representative_map_idempotent {A : Type} {R : A → A → Prop} {f : A → A}
+  (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b ↔ R a a ∧ R b b ∧ f a = f b) (a : A)
+  : f (f a) = f a :=
+symm (and_elim_right (and_elim_right (iff_elim_left (H2 a (f a)) (H1 a))))
+
+theorem representative_map_idempotent_equiv {A : Type} {R : A → A → Prop} {f : A → A}
+  (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b) (a : A)
+  : f (f a) = f a :=
+symm (H2 a (f a) (H1 a))
+
+theorem representative_map_refl_rep {A : Type} {R : A → A → Prop} {f : A → A}
+  (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b ↔ R a a ∧ R b b ∧ f a = f b) (a : A)
+  : R (f a) (f a) :=
+subst (representative_map_idempotent H1 H2 a) (H1 (f a)) 
+
+theorem representative_map_image_fix {A : Type} {R : A → A → Prop} {f : A → A}
+  (H1 : ∀a, R a (f a)) (H2 : ∀a a', R a a' ↔ R a a ∧ R a' a' ∧ f a = f a') (b : image f)
+  : f (elt_of b) = elt_of b :=
+idempotent_image_fix (representative_map_idempotent H1 H2) b
+
+theorem representative_map_to_quotient {A : Type} {R : A → A → Prop} {f : A → A}
+  (H1 : ∀a, R a (f a)) (H2 : ∀a a', R a a' ↔ R a a ∧ R a' a' ∧ f a = f a')
+  : is_quotient _ (fun_image f) elt_of :=
+let abs [inline] := fun_image f in
+intro
+  (take u : image f,
+    obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
+    have H : elt_of (abs (elt_of u)) = elt_of u, from
+      calc
+        elt_of (abs (elt_of u)) = f (elt_of u) : elt_of_fun_image _ _
+          ... = f (f a) : {symm Ha}
+          ... = f a : representative_map_idempotent H1 H2 a
+          ... = elt_of u : Ha,
+    show abs (elt_of u) = u, from subtype_eq H)
+  (take u : image f,
+    show R (elt_of u) (elt_of u), from
+      obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
+        subst Ha (@representative_map_refl_rep A R f H1 H2 a))
+  (take a a', 
+    substi (fun_image_eq f a a') (H2 a a'))
+
+-- TODO: fix these
+-- e.g. in the next three lemmas, we should not need to specify the equivalence relation
+-- but the class inference finds reflexive.class eq
+theorem equiv_is_refl {A : Type} {R : A → A → Prop} (equiv : is_equivalence.class R) :=
+@operations.refl _ R (@is_equivalence.is_reflexive _ _ equiv)
+-- we should be able to write
+-- @operations.refl _ R _
+
+theorem equiv_is_symm {A : Type} {R : A → A → Prop} (equiv : is_equivalence.class R) :=
+@operations.symm _ R (@is_equivalence.is_symmetric _ _ equiv)
+
+theorem equiv_is_trans {A : Type} {R : A → A → Prop} (equiv : is_equivalence.class R) :=
+@operations.trans _ R (@is_equivalence.is_transitive _ _ equiv)
+
+theorem representative_map_equiv_inj {A : Type} {R : A → A → Prop}
+  (equiv : is_equivalence.class R) {f : A → A} (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b)
+  {a b : A} (H3 : f a = f b) : R a b :=
+-- have symmR : symmetric R, from @relation.operations.symm _ R _,
+have symmR : symmetric R, from equiv_is_symm equiv,
+have transR : transitive R, from equiv_is_trans equiv,
+show R a b, from
+  have H2 : R a (f b), from subst H3 (H1 a),
+  have H3 : R (f b) b, from symmR _ _ (H1 b),
+  transR _ _ _ H2 H3
+
+theorem representative_map_to_quotient_equiv {A : Type} {R : A → A → Prop}
+  (equiv : is_equivalence.class R) {f : A → A} (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b)
+  : @is_quotient A (image f) R (fun_image f) elt_of :=
+representative_map_to_quotient
+  H1
+  (take a b,
+    have reflR : reflexive R, from equiv_is_refl equiv,
+    have H3 : f a = f b → R a b, from representative_map_equiv_inj equiv H1 H2,
+    have H4 : R a b ↔ f a = f b, from iff_intro (H2 a b) H3,
+    have H5 : R a b ↔ R b b ∧ f a = f b,
+      from substi (symmi (and_inhabited_left _ (reflR b))) H4,
+    substi (symmi (and_inhabited_left _ (reflR a))) H5)
+
+-- TODO: split this into another file -- it depends on hilbert
+
+-- abstract quotient
+-- -----------------
+
+definition prelim_map {A : Type} (R : A → A → Prop) (a : A) :=
+-- TODO: it is interesting how the elaborator fails here
+-- epsilon (fun b, R a b)
+@epsilon _ (nonempty_intro a) (fun b, R a b)
+
+-- TODO: only needed R reflexive (or weaker: R a a)
+theorem prelim_map_rel {A : Type} {R : A → A → Prop} (H : is_equivalence.class R) (a : A)
+  : R a (prelim_map R a) :=
+have reflR : reflexive R, from equiv_is_refl H,
+epsilon_spec (exists_intro a (reflR a))
+
+-- TODO: only needed: R PER
+theorem prelim_map_congr {A : Type} {R : A → A → Prop} (H1 : is_equivalence.class R) {a b : A}
+  (H2 : R a b) : prelim_map R a = prelim_map R b :=
+have symmR : symmetric R, from equiv_is_symm H1,
+have transR : transitive R, from equiv_is_trans H1,
+have H3 : ∀c, R a c ↔ R b c, from
+  take c,
+    iff_intro
+      (assume H4 : R a c, transR b a c (symmR a b H2) H4)
+      (assume H4 : R b c, transR a b c H2 H4),
+have H4 : (fun c, R a c) = (fun c, R b c), from funext (take c, iff_to_eq (H3 c)),
+show @epsilon _ (nonempty_intro a) (λc, R a c) = @epsilon _ (nonempty_intro b) (λc, R b c), 
+  from congr2 _ H4
+
+definition quotient {A : Type} (R : A → A → Prop) : Type := image (prelim_map R)
+
+definition quotient_abs {A : Type} (R : A → A → Prop) : A → quotient R :=
+fun_image (prelim_map R)
+
+definition quotient_elt_of {A : Type} (R : A → A → Prop) : quotient R → A := elt_of
+
+theorem quotient_is_quotient  {A : Type} (R : A → A → Prop) (H : is_equivalence.class R)
+  : is_quotient R (quotient_abs R) (quotient_elt_of R) :=
+representative_map_to_quotient_equiv
+  H
+  (prelim_map_rel H)
+  (@prelim_map_congr _ _ H)
+
+-- previously:
+-- opaque_hint (hiding fun_image rec is_quotient prelim_map)
+-- transparent: image, image_incl, quotient, quotient_abs, quotient_rep
+
+end quotient

library/standard/data/sigma.lean

@@ -38,7 +38,7 @@ section
 	assume (H2' : eq_rec_on H1' b1 = b2'),
 	show dpair a1 b1 = dpair a1 b2', from
 	  calc
-	    dpair a1 b1 = dpair a1 (eq_rec_on H1' b1) : {symm (eq_rec_on_irrel H1' b1)}
+	    dpair a1 b1 = dpair a1 (eq_rec_on H1' b1) : {symm (eq_rec_on_id H1' b1)}
 	      ... = dpair a1 b2' : {H2'}) H1)
   b2 H1 H2
 

library/standard/data/subtype.lean

@@ -27,7 +27,7 @@ section
 
   theorem tag_irrelevant {a : A} (H1 H2 : P a) : tag a H1 = tag a H2 := refl (tag a H1)
 
-  theorem tag_ext (a : subtype P) : Π(H : P (elt_of a)), tag (elt_of a) H = a :=
+  theorem tag_elt_of (a : subtype P) : Π(H : P (elt_of a)), tag (elt_of a) H = a :=
     subtype_destruct a (take (x : A) (H1 : P x) (H2 : P x), refl _)
 
   theorem tag_eq {a1 a2 : A} {H1 : P a1} {H2 : P a2} (H3 : a1 = a2) : tag a1 H1 = tag a2 H2 :=

library/standard/logic/classes/classes.md

@@ -4,5 +4,13 @@ logic.classes
 Useful classes for general logical manipulations.
 
 * [inhabited](inhabited.lean) : inhabited types
+* [nonempty](nonempty.lean) : nonempty type
 * [decidable](decidable.lean) : decidable types
 * [congr](congr.lean) : congruences with respect to suitable relations
+
+Constructively, inhabited types have a witness, while nonempty types
+are "proof irrelevant". Classically (assuming the axiom in
+`logic.axioms.hilbert`) the two are equivalent. Type class inferences
+are set up to use "inhabited" however, so users should use that to
+declare that types have an element. Use "nonempty" in the hypothesis
+of a theorem when the theorem does not depend on the witness chosen.

library/standard/logic/connectives/eq.lean

@@ -14,7 +14,12 @@ inductive eq {A : Type} (a : A) : A → Prop :=
 
 infix `=`:50 := eq
 
-theorem eq_irrel {A : Type} {a : A} (H1 : a = a) : H1 = (refl a) := refl _
+-- TODO: try this out -- shorthand for "refl _"
+notation `rfl`:max := refl _
+
+theorem eq_id_refl {A : Type} {a : A} (H1 : a = a) : H1 = (refl a) := rfl
+
+theorem eq_irrel {A : Type} {a b : A} (H1 H2 : a = b) : H1 = H2 := rfl
 
 theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
 eq_rec H2 H1
@@ -37,11 +42,17 @@ assume H : true = false,
 definition eq_rec_on {A : Type} {a1 a2 : A} {B : A → Type} (H1 : a1 = a2) (H2 : B a1) : B a2 :=
 eq_rec H2 H1
 
-theorem eq_rec_on_irrel {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : eq_rec_on H b = b :=
+theorem eq_rec_on_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : eq_rec_on H b = b :=
 @trans _ _ (eq_rec_on (refl a) b) _ (refl _) (refl _)
 
-theorem eq_rec_irrel {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : eq_rec b H = b :=
+theorem eq_rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : eq_rec b H = b :=
-eq_rec_on_irrel H b
+eq_rec_on_id H b
+
+theorem eq_rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (H2 : b = c) (u : P a) :
+  eq_rec_on H2 (eq_rec_on H1 u) = eq_rec_on (trans H1 H2) u :=
+(show ∀(H2 : b = c), eq_rec_on H2 (eq_rec_on H1 u) = eq_rec_on (trans H1 H2) u,
+  from eq_rec_on H2 (take (H2 : b = b), eq_rec_on_id H2 _))
+H2
 
 namespace eq_proofs
   postfix `⁻¹`:100 := symm

library/standard/struc/relation.lean

@@ -1,8 +1,6 @@
-----------------------------------------------------------------------------------------------------
 -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Jeremy Avigad
-----------------------------------------------------------------------------------------------------
 
 import logic.connectives.prop
 
@@ -20,11 +18,11 @@ namespace is_reflexive
   inductive class {T : Type} (R : T → T → Type) : Prop :=
   | mk : reflexive R → class R
 
-abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) : reflexive R
+  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) : reflexive R :=
-:= class_rec (λu, u) C
+  class_rec (λu, u) C
 
-abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} : reflexive R
+  abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} : reflexive R :=
-:= class_rec (λu, u) C
+  class_rec (λu, u) C
 
 end is_reflexive
 
@@ -33,11 +31,11 @@ namespace is_symmetric
   inductive class {T : Type} (R : T → T → Type) : Prop :=
   | mk : symmetric R → class R
 
-abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) ⦃x y : T⦄ (H : R x y) : R y x
+  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) ⦃x y : T⦄ (H : R x y) : R y x :=
-:= class_rec (λu, u) C x y H
+  class_rec (λu, u) C x y H
 
-abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} ⦃x y : T⦄ (H : R x y) : R y x
+  abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} ⦃x y : T⦄ (H : R x y) : R y x :=
-:= class_rec (λu, u) C x y H
+  class_rec (λu, u) C x y H
 
 end is_symmetric
 
@@ -47,15 +45,51 @@ inductive class {T : Type} (R : T → T → Type)  : Prop :=
   | mk : transitive R → class R
 
   abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) ⦃x y z : T⦄ (H1 : R x y)
-  (H2 : R y z) : R x z
+    (H2 : R y z) : R x z :=
-:= class_rec (λu, u) C x y z H1 H2
+  class_rec (λu, u) C x y z H1 H2
 
   abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} ⦃x y z : T⦄ (H1 : R x y)
-  (H2 : R y z) : R x z
+    (H2 : R y z) : R x z :=
-:= class_rec (λu, u) C x y z H1 H2
+  class_rec (λu, u) C x y z H1 H2
 
 end is_transitive
 
+namespace is_equivalence
+
+  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  | mk : is_reflexive.class R → is_symmetric.class R → is_transitive.class R → class R
+
+  theorem is_reflexive {T : Type} {R : T → T → Type} {C : class R} : is_reflexive.class R :=
+  class_rec (λx y z, x) C
+
+  theorem is_symmetric {T : Type} {R : T → T → Type} {C : class R} : is_symmetric.class R :=
+  class_rec (λx y z, y) C
+
+  theorem is_transitive {T : Type} {R : T → T → Type} {C : class R} : is_transitive.class R :=
+  class_rec (λx y z, z) C
+
+end is_equivalence
+
+instance is_equivalence.is_reflexive
+instance is_equivalence.is_symmetric
+instance is_equivalence.is_transitive
+
+namespace is_PER
+
+  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  | mk : is_symmetric.class R → is_transitive.class R → class R
+
+  theorem is_symmetric {T : Type} {R : T → T → Type} {C : class R} : is_symmetric.class R :=
+  class_rec (λx y, x) C
+
+  theorem is_transitive {T : Type} {R : T → T → Type} {C : class R} : is_transitive.class R :=
+  class_rec (λx y, y) C
+
+end is_PER
+
+-- instance is_PER.is_symmetric
+instance is_PER.is_transitive
+
 
 -- Congruence for unary and binary functions
 -- -----------------------------------------