feat(library/standard): port int, and reorganize a lot

library/standard/data/bool.lean

@@ -2,8 +2,12 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Leonardo de Moura
 
-import .type logic.connectives.basic logic.classes.decidable logic.classes.inhabited
+import logic.connectives.basic logic.classes.decidable logic.classes.inhabited
-using eq_proofs decidable
+using eq_ops decidable
+
+inductive bool : Type :=
+| ff : bool
+| tt : bool
 
 namespace bool
 
@@ -11,7 +15,7 @@ theorem induction_on {p : bool → Prop} (b : bool) (H0 : p ff) (H1 : p tt) : p
 bool_rec H0 H1 b
 
 theorem inhabited_bool [instance] : inhabited bool :=
-inhabited_intro ff
+inhabited_mk ff
 
 definition cond {A : Type} (b : bool) (t e : A) :=
 bool_rec e t b
@@ -136,4 +140,4 @@ theorem bnot_false : !ff = tt := refl _
 
 theorem bnot_true  : !tt = ff := refl _
 
-end bool
+end bool

library/standard/data/bool/bool.md

@@ -1,7 +0,0 @@
-data.bool
-=========
-
-The type of booleans.
-
-* [type](type.lean) : the datatype
-* [basic](basic.lean) : basic properties

library/standard/data/bool/type.lean

@@ -1,11 +0,0 @@
--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Leonardo de Moura
-
-namespace bool
-
-inductive bool : Type :=
-| ff : bool
-| tt : bool
-
-end bool

library/standard/data/data.md

@@ -7,7 +7,7 @@ Basic types:
 
 * [empty](empty.lean) : the empty type
 * [unit](unit.lean) : the singleton type
-* [bool](bool/bool.md) : the boolean values
+* [bool](bool.lean) : the boolean values
 * [num](num.lean) : generic numerals
 * [string](string.lean) : ascii strings
 * [nat](nat/nat.md) : the natural numbers
@@ -20,5 +20,6 @@ Constructors:
 * [sigma](sigma.lean) : the dependent product
 * [option](option.lean)
 * [subtype](subtype.lean)
+* [quotient](quotient/quotient.md)
 * [list](list/list.md)
 * [set](set.lean)

library/standard/data/default.lean

@@ -0,0 +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 .empty .unit .bool .num .string .nat .int
+import .prod .sum .sigma .option .subtype .quotient .list .set

library/standard/data/int/default.lean

@@ -2,4 +2,4 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Jeremy Avigad
 
-import .type .basic
+import data.nat.basic

library/standard/data/int/int.default

@@ -0,0 +1,5 @@
+--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+--- Released under Apache 2.0 license as described in the file LICENSE.
+--- Author: Jeremy Avigad
+
+import .basic

library/standard/data/int/int.md

@@ -0,0 +1,6 @@
+data.int
+========
+
+The integers.
+
+* [basic](basic.lean) : the integers, with basic operations and order.

library/standard/data/list/basic.lean

@@ -15,7 +15,7 @@ import logic
 -- import if -- for find
 
 using nat
-using eq_proofs
+using eq_ops
 
 namespace list
 

library/standard/data/nat/basic.lean

@@ -1,12 +1,11 @@
-----------------------------------------------------------------------------------------------------
 --- 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 data.num tools.tactic struc.binary
-using num tactic binary eq_proofs
+using num tactic binary eq_ops
 using decidable (hiding induction_on rec_on)
+using relation -- for subst_iff
 
 -- TODO: this should go in tools, I think
 namespace helper_tactics
@@ -29,7 +28,19 @@ inductive nat : Type :=
 | zero : nat
 | succ : nat → nat
 
-notation `ℕ` : max := nat
+notation `ℕ`:max := nat
+
+theorem nat_rec_zero {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) : nat_rec x f zero = x
+
+theorem nat_rec_succ {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) (n : ℕ) :
+  nat_rec x f (succ n) = f n (nat_rec x f n)
+
+theorem induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P zero) (H2 : ∀ (n : ℕ) (IH : P n), P (succ n)) :
+  P a :=
+nat_rec H1 H2 a
+
+definition rec_on {P : ℕ → Type} (n : ℕ) (H1 : P zero) (H2 : ∀m, P m → P (succ m)) : P n :=
+nat_rec H1 H2 n
 
 
 -- Coercion from num
@@ -42,30 +53,20 @@ definition to_nat [coercion] [inline] (n : num) : ℕ :=
 num_rec zero
     (λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
 
-theorem nat_rec_zero {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat_rec x f 0 = x
-
-theorem nat_rec_succ {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) (n : ℕ) :
-  nat_rec x f (succ n) = f n (nat_rec x f n)
-
-theorem induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : ∀ (n : ℕ) (IH : P n), P (succ n)) :
-  P a :=
-nat_rec H1 H2 a
-
-definition rec_on {P : ℕ → Type} (n : ℕ) (H1 : P 0) (H2 : ∀m, P m → P (succ m)) : P n :=
-nat_rec H1 H2 n
-
 
 -- Successor and predecessor
 -- -------------------------
 
+-- TODO: this looks like a calc bug -- calc is using subst for iff, instead of =
+calc_subst subst
 theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
 assume H : succ n = 0,
 have H2 : true = false, from
   let f [inline] := (nat_rec false (fun a b, true)) in
     calc
-      true = f (succ n) : _
+      true = f (succ n) : rfl
        ... = f 0        : {H}
-       ... = false      : _,
+       ... = false      : rfl,
 absurd H2 true_ne_false
 
 -- add_rewrite succ_ne_zero
@@ -82,7 +83,7 @@ theorem zero_or_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
 induction_on n
   (or_inl (refl 0))
   (take m IH, or_inr
-    (show succ m = succ (pred (succ m)), from congr2 succ (pred_succ m⁻¹)))
+    (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
 
 theorem zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
 or_imp_or (zero_or_succ_pred n) (assume H, H)
@@ -122,7 +123,7 @@ have general : ∀n, decidable (n = m), from
 	(λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
 	  have d1 : decidable (n' = m'), from iH1 n',
 	  decidable.rec_on d1
-	    (assume Heq : n' = m', inl (congr2 succ Heq))
+	    (assume Heq : n' = m', inl (congr_arg succ Heq))
 	    (assume Hne : n' ≠ m',
 	      have H1 : succ n' ≠ succ m', from
 		assume Heq, absurd (succ_inj Heq) Hne,
@@ -415,4 +416,4 @@ discriminate
 -- add_rewrite mul_succ_left mul_succ_right
 -- add_rewrite mul_comm mul_assoc mul_left_comm
 -- add_rewrite mul_distr_right mul_distr_left
-end nat
+end nat

library/standard/data/nat/order.lean

@@ -1,14 +1,12 @@
-----------------------------------------------------------------------------------------------------
 --- 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 .basic
-using nat eq_proofs tactic
+import tools.fake_simplifier
 
--- until we have the simplifier...
+using nat eq_ops tactic
-definition simp : tactic := apply @sorry
+using fake_simplifier
 
 -- TODO: move these to logic.connectives
 theorem or_imp_or_left {a b c : Prop} (H1 : a ∨ b) (H2 : a → c) : c ∨ b :=
@@ -30,7 +28,6 @@ namespace nat
 definition le (n m : ℕ) : Prop := exists k : nat, n + k = m
 
 infix  `<=`:50 := le
-
 infix  `≤`:50  := le
 
 theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m :=
@@ -583,4 +580,4 @@ mul_eq_one_left ((mul_comm n m) ▸ H)
 
 --- theorem mul_eq_one {n m : ℕ} (H : n * m = 1) : n = 1 ∧ m = 1
 --- := and_intro (mul_eq_one_left H) (mul_eq_one_right H)
-end nat
+end nat

library/standard/data/nat/sub.lean

@@ -1,13 +1,15 @@
-----------------------------------------------------------------------------------------------------
 --- 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 data.nat.order
-using nat eq_proofs tactic
+import tools.fake_simplifier
+
+using nat eq_ops tactic
 using helper_tactics
 
+using fake_simplifier
+
 namespace nat
 
 -- data.nat.basic2
@@ -508,4 +510,4 @@ or_elim (le_total k l)
   (assume H : k ≤ l, dist_comm l k ▸ dist_comm _ _ ▸ aux l k H)
   (assume H : l ≤ k, aux k l H)
 
-end nat
+end nat

library/standard/data/num.lean

@@ -21,9 +21,9 @@ inductive num : Type :=
 | pos   : pos_num → num
 
 theorem inhabited_pos_num [instance] : inhabited pos_num :=
-inhabited_intro one
+inhabited_mk one
 
 theorem inhabited_num [instance] : inhabited num :=
-inhabited_intro zero
+inhabited_mk zero
 
-end num
+end num

library/standard/data/option.lean

@@ -4,7 +4,7 @@
 -- Author: Leonardo de Moura
 ----------------------------------------------------------------------------------------------------
 import logic.connectives.basic logic.connectives.eq logic.classes.inhabited logic.classes.decidable
-using eq_proofs decidable
+using eq_ops decidable
 
 namespace option
 inductive option (A : Type) : Type :=
@@ -32,10 +32,10 @@ assume H : none = some a, absurd
   (not_is_none_some a)
 
 theorem some_inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
-congr2 (option_rec a₁ (λ a, a)) H
+congr_arg (option_rec a₁ (λ a, a)) H
 
 theorem inhabited_option [instance] (A : Type) : inhabited (option A) :=
-inhabited_intro none
+inhabited_mk none
 
 theorem decidable_eq [instance] {A : Type} {H : ∀a₁ a₂ : A, decidable (a₁ = a₂)} (o₁ o₂ : option A) : decidable (o₁ = o₂) :=
 rec_on o₁
@@ -45,4 +45,4 @@ rec_on o₁
     (take a₂ : A, decidable.rec_on (H a₁ a₂)
       (assume Heq : a₁ = a₂, inl (Heq ▸ refl _))
       (assume Hne : a₁ ≠ a₂, inr (assume Hn : some a₁ = some a₂, absurd (some_inj Hn) Hne))))
-end option
+end option

library/standard/data/prod.lean

@@ -4,6 +4,8 @@
 
 import logic.classes.inhabited logic.connectives.eq
 
+using inhabited
+
 inductive prod (A B : Type) : Type :=
 | pair : A → B → prod A B
 
@@ -39,10 +41,10 @@ section
   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)))
+  inhabited_destruct H1 (λa, inhabited_destruct H2 (λb, inhabited_mk (pair a b)))
 
 end
 
 instance prod_inhabited
 
-end prod
+end prod

library/standard/data/quotient/aux.lean

@@ -0,0 +1,175 @@
+-- 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 ..prod struc.relation
+import tools.fake_simplifier
+
+using prod eq_ops
+using fake_simplifier
+
+-- TODO: calc bug -- remove
+calc_subst subst
+
+
+namespace quotient
+
+-- auxliary facts about products
+-- -----------------------------
+
+-- ### 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 congr_arg 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)
+
+end quotient

library/standard/data/quotient/basic.lean

@@ -2,197 +2,11 @@
 -- 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 tools.tactic ..subtype logic.connectives.cast struc.relation data.prod
 import logic.connectives.instances
+import .aux
 
--- for now: to use substitution (iff_to_eq)
+using relation prod inhabited nonempty tactic eq_ops
-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
 -- ====================
@@ -205,7 +19,7 @@ using subtype
 -- ---------------------
 
 -- TODO: make this a structure
-definition is_quotient {A B : Type} (R : A → A → Prop) (abs : A → B) (rep : B → A) : Prop :=
+abbreviation 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))
@@ -224,8 +38,8 @@ and_intro H1 (and_intro H2 H3)
 --   (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_absorb_left _ (H1 s))) (H3 r s),
---     gensubst.subst (relation.operations.symm (and_inhabited_left _ (H1 r))) H4)
+--     gensubst.subst (relation.operations.symm (and_absorb_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 :=
@@ -233,7 +47,7 @@ and_intro H1 (and_intro H2 H3)
 
 -- 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)
+-- relation.operations.symm (and_absorb_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) :=
@@ -241,7 +55,10 @@ and_intro H1 (and_intro H2 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
+-- gensubst.subst (relation.operations.symm (and_absorb_left Q H1)) H3
+
+theorem and_absorb_left {a : Prop} (b : Prop) (Ha : a) : a ∧ b ↔ b :=
+iff_intro (assume Hab, and_elim_right Hab) (assume Hb, and_intro Ha Hb)
 
 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)
@@ -252,8 +69,20 @@ intro
   (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),
+        subst_iff (iff_symm (and_absorb_left _ (H1 s))) (H3 r s),
-    substi (symmi (and_inhabited_left _ (H1 r))) H4)
+    subst_iff (iff_symm (and_absorb_left _ (H1 r))) H4)
+
+-- 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 (iff_symm (and_absorb_left _ (H1 s))) (H3 r s),
+--     substi (iff_symm (and_absorb_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 := 
@@ -414,10 +243,10 @@ opaque_hint (hiding rec rec_constant rec_binary quotient_map quotient_map_binary
 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))
+inhabited_mk (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)
+image_inhabited f (inhabited_mk a)
 
 definition fun_image {A B : Type} (f : A → B) (a : A) : image f :=
 tag (f a) (exists_intro a rfl)
@@ -446,7 +275,7 @@ theorem fun_image_eq {A B : Type} (f : A → B) (a a' : 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'))
+    subst (subst (congr_arg 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 :=
@@ -508,89 +337,30 @@ intro
       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'))
+    subst_iff (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)
+  (equiv : is_equivalence 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 is_symmetric.infer R,
-have symmR : symmetric R, from equiv_is_symm equiv,
+have transR : transitive R, from is_transitive.infer R,
-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),
+  have H3 : R (f b) b, from symmR (H1 b),
-  transR _ _ _ H2 H3
+  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)
+  (equiv : is_equivalence 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 reflR : reflexive R, from is_reflexive.infer R,
     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,
+      from subst_iff (iff_symm (and_absorb_left _ (reflR b))) H4,
-    substi (symmi (and_inhabited_left _ (reflR a))) H5)
+    subst_iff (iff_symm (and_absorb_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)

library/standard/data/quotient/classical.lean

@@ -0,0 +1,56 @@
+-- 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 struc.relation logic.classes.nonempty data.subtype
+import .basic
+import logic.axioms.classical logic.axioms.hilbert logic.axioms.funext
+
+namespace quotient
+
+using relation nonempty subtype
+
+-- 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 R) (a : A)
+  : R a (prelim_map R a) :=
+have reflR : reflexive R, from is_reflexive.infer R,
+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 R) {a b : A}
+  (H2 : R a b) : prelim_map R a = prelim_map R b :=
+have symmR : symmetric R, from is_symmetric.infer R,
+have transR : transitive R, from is_transitive.infer R,
+have H3 : ∀c, R a c ↔ R b c, from
+  take c,
+    iff_intro
+      (assume H4 : R a c, transR (symmR H2) H4)
+      (assume H4 : R b c, transR 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 congr_arg _ 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
+
+-- TODO: I had to make is_quotient transparent -- change this?
+theorem quotient_is_quotient  {A : Type} (R : A → A → Prop) (H : is_equivalence 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)
+
+end quotient

library/standard/data/quotient/default.lean

@@ -0,0 +1,5 @@
+--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+--- Released under Apache 2.0 license as described in the file LICENSE.
+--- Author: Jeremy Avigad
+
+import .basic .classical

library/standard/data/quotient/quotient.md

@@ -0,0 +1,6 @@
+data.quotient
+=============
+
+* [aux](aux.lean) : auxiliary facts about products
+* [basic](basic.lean) : the constructive core of the quotient construction
+* [classical](classical.lean) : the classical version, using Hilbert choice

library/standard/data/set.lean

@@ -5,7 +5,7 @@
 ----------------------------------------------------------------------------------------------------
 
 import logic.axioms.funext data.bool
-using eq_proofs bool
+using eq_ops bool
 
 namespace set
 definition set (T : Type) := T → bool

library/standard/data/sigma.lean

@@ -4,6 +4,8 @@
 
 import logic.classes.inhabited logic.connectives.eq
 
+using inhabited
+
 inductive sigma {A : Type} (B : A → Type) : Type :=
 | dpair : Πx : A, B x → sigma B
 
@@ -48,7 +50,7 @@ section
 
   theorem sigma_inhabited (H1 : inhabited A) (H2 : inhabited (B (default A))) :
     inhabited (sigma B) :=
-   inhabited_elim H1 (λa, inhabited_elim H2 (λb, inhabited_intro (dpair (default A) b)))
+   inhabited_destruct H1 (λa, inhabited_destruct H2 (λb, inhabited_mk (dpair (default A) b)))
 
 end
 

library/standard/data/string.lean

@@ -1,13 +1,13 @@
-----------------------------------------------------------------------------------------------------
 -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Leonardo de Moura
-----------------------------------------------------------------------------------------------------
 
 import data.bool
-using bool
+
+using bool inhabited
 
 namespace string
+
 inductive char : Type :=
 | ascii : bool → bool → bool → bool → bool → bool → bool → bool → char
 
@@ -16,9 +16,9 @@ inductive string : Type :=
 | str   : char → string → string
 
 theorem inhabited_char [instance] : inhabited char :=
-inhabited_intro (ascii ff ff ff ff ff ff ff ff)
+inhabited_mk (ascii ff ff ff ff ff ff ff ff)
 
 theorem inhabited_string [instance] : inhabited string :=
-inhabited_intro empty
+inhabited_mk empty
 
 end string

library/standard/data/unit.lean

@@ -17,8 +17,8 @@ theorem unit_eq (a b : unit) : a = b :=
 unit_rec (unit_rec (refl ⋆) b) a
 
 theorem inhabited_unit [instance] : inhabited unit :=
-inhabited_intro ⋆
+inhabited_mk ⋆
 
 theorem decidable_eq [instance] (a b : unit) : decidable (a = b) :=
 inl (unit_eq a b)
-end unit
+end unit

library/standard/hott/inhabited.lean

@@ -1,40 +0,0 @@
--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Leonardo de Moura
-
--- The predicative version of inhabited
--- TODO: restore instances
-
--- import logic bool
--- using logic
-
-namespace predicative
-
-inductive inhabited (A : Type) : Type :=
-| inhabited_intro : A → inhabited A
-
-theorem inhabited_elim {A : Type} {B : Type} (H1 : inhabited A) (H2 : A → B) : B
-:= inhabited_rec H2 H1
-
-end predicative
-
--- theorem inhabited_fun [instance] (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B)
--- := inhabited_elim H (take (b : B), inhabited_intro (λ a : A, b))
-
--- theorem inhabited_sum_left [instance] {A : Type} (B : Type) (H : inhabited A) : inhabited (A + B)
--- := inhabited_elim H (λ a, inhabited_intro (inl B a))
-
--- theorem inhabited_sum_right [instance] (A : Type) {B : Type} (H : inhabited B) : inhabited (A + B)
--- := inhabited_elim H (λ b, inhabited_intro (inr A b))
-
--- theorem inhabited_product [instance] {A : Type} {B : Type} (Ha : inhabited A) (Hb : inhabited B) : inhabited (A × B)
--- := inhabited_elim Ha (λ a, (inhabited_elim Hb (λ b, inhabited_intro (a, b))))
-
--- theorem inhabited_bool [instance] : inhabited bool
--- := inhabited_intro true
-
--- theorem inhabited_unit [instance] : inhabited unit
--- := inhabited_intro ⋆
-
--- theorem inhabited_sigma_pr1 {A : Type} {B : A → Type} (p : Σ x, B x) : inhabited A
--- := inhabited_intro (dpr1 p)

library/standard/logic/axioms/classical.lean

@@ -4,9 +4,9 @@
 -- Author: Leonardo de Moura
 ----------------------------------------------------------------------------------------------------
 
-import logic.connectives.basic logic.connectives.quantifiers logic.connectives.cast
+import logic.connectives.basic logic.connectives.quantifiers logic.connectives.cast struc.relation
 
-using eq_proofs
+using eq_ops
 
 axiom prop_complete (a : Prop) : a = true ∨ a = false
 
@@ -15,6 +15,9 @@ or_elim (prop_complete a)
   (assume Ht : a = true,  Ht⁻¹ ▸ H1)
   (assume Hf : a = false, Hf⁻¹ ▸ H2)
 
+theorem case_on (a : Prop) {P : Prop → Prop} (H1 : P true) (H2 : P false) : P a :=
+case P H1 H2 a
+
 theorem em (a : Prop) : a ∨ ¬a :=
 or_elim (prop_complete a)
   (assume Ht : a = true,  or_inl (eqt_elim Ht))
@@ -163,3 +166,12 @@ theorem peirce (a b : Prop) : ((a → b) → a) → a :=
 assume H, by_contradiction (assume Hna : ¬a,
   have Hnna : ¬¬a, from not_implies_left (mt H Hna),
   absurd (not_not_elim Hnna) Hna)
+
+-- with classical logic, every predicate respects iff
+
+using relation
+theorem iff_congr [instance] (P : Prop → Prop) : congr iff iff P :=
+congr_mk
+  (take (a b : Prop),
+    assume H : a ↔ b,
+    show P a ↔ P b, from eq_to_iff (subst (iff_to_eq H) (refl (P a))))

library/standard/logic/axioms/examples/diaconescu.lean

@@ -3,7 +3,7 @@
 -- Author: Leonardo de Moura
 
 import logic.axioms.hilbert logic.axioms.funext
-using eq_proofs
+using eq_ops nonempty inhabited
 
 -- Diaconescu’s theorem
 -- Show that Excluded middle follows from

library/standard/logic/axioms/hilbert.lean

@@ -6,7 +6,7 @@ import logic.connectives.eq logic.connectives.quantifiers
 import logic.classes.inhabited logic.classes.nonempty
 import data.subtype data.sum
 
-using subtype
+using subtype inhabited nonempty
 
 -- logic.axioms.hilbert
 -- ====================
@@ -25,7 +25,7 @@ nonempty_elim H (take x, exists_intro x trivial)
 
 theorem nonempty_imp_inhabited {A : Type} (H : nonempty A) : inhabited A :=
 let u : {x : A | (∃x : A, true) → true} := strong_indefinite_description (λa, true) H in
-inhabited_intro (elt_of u)
+inhabited_mk (elt_of u)
 
 theorem inhabited_exists {A : Type} {P : A → Prop} (H : ∃x, P x) : inhabited A :=
 nonempty_imp_inhabited (obtain w Hw, from H, nonempty_intro w)

library/standard/logic/axioms/piext.lean

@@ -6,27 +6,29 @@
 
 import logic.classes.inhabited logic.connectives.cast
 
+using inhabited
+
 -- Pi extensionality
 axiom piext {A : Type} {B B' : A → Type} {H : inhabited (Π x, B x)} :
   (Π x, B x) = (Π x, B' x) → B = B'
 
 theorem cast_app {A : Type} {B B' : A → Type} (H : (Π x, B x) = (Π x, B' x)) (f : Π x, B x)
   (a : A) : cast H f a == f a :=
-have Hi [fact] : inhabited (Π x, B x), from inhabited_intro f,
+have Hi [fact] : inhabited (Π x, B x), from inhabited_mk f,
 have Hb : B = B', from piext H,
 cast_app' Hb f a
 
-theorem hcongr1 {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
+theorem hcongr_fun {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
   (H : f == f') : f a == f' a :=
-have Hi [fact] : inhabited (Π x, B x), from inhabited_intro f,
+have Hi [fact] : inhabited (Π x, B x), from inhabited_mk f,
 have Hb : B = B', from piext (type_eq H),
-hcongr1' a H Hb
+hcongr_fun' a H Hb
 
 theorem hcongr {A A' : Type} {B : A → Type} {B' : A' → Type}
     {f : Π x, B x} {f' : Π x, B' x} {a : A} {a' : A'}
     (Hff' : f == f') (Haa' : a == a') : f a == f' a' :=
 have H1 : ∀ (B B' : A → Type) (f : Π x, B x) (f' : Π x, B' x), f == f' → f a == f' a, from
-  take B B' f f' e, hcongr1 a e,
+  take B B' f f' e, hcongr_fun a e,
 have H2 : ∀ (B : A → Type) (B' : A' → Type) (f : Π x, B x) (f' : Π x, B' x),
     f == f' → f a == f' a', from hsubst Haa' H1,
 H2 B B' f f' Hff'

library/standard/logic/axioms/prop_decidable.lean

@@ -5,7 +5,7 @@
 ----------------------------------------------------------------------------------------------------
 
 import logic.axioms.classical logic.axioms.hilbert logic.classes.decidable
-using decidable
+using decidable inhabited nonempty
 
 -- Excluded middle + Hilbert implies every proposition is decidable
 

library/standard/logic/classes/decidable.lean

@@ -27,12 +27,12 @@ decidable_rec H1 H2 H
 theorem irrelevant {p : Prop} (d1 d2 : decidable p) : d1 = d2 :=
 decidable_rec
   (assume Hp1 : p, decidable_rec
-    (assume Hp2  : p,  congr2 inl (refl Hp1)) -- using proof irrelevance for Prop
+    (assume Hp2  : p,  congr_arg inl (refl Hp1)) -- using proof irrelevance for Prop
     (assume Hnp2 : ¬p, absurd_elim (inl Hp1 = inr Hnp2) Hp1 Hnp2)
     d2)
   (assume Hnp1 : ¬p, decidable_rec
     (assume Hp2  : p,  absurd_elim (inr Hnp1 = inl Hp2) Hp2 Hnp1)
-    (assume Hnp2 : ¬p, congr2 inr (refl Hnp1)) -- using proof irrelevance for Prop
+    (assume Hnp2 : ¬p, congr_arg inr (refl Hnp1)) -- using proof irrelevance for Prop
     d2)
   d1
 
@@ -87,4 +87,4 @@ rec_on Ha
 theorem decidable_eq_equiv {a b : Prop} (Ha : decidable a) (H : a = b) : decidable b :=
 decidable_iff_equiv Ha (eq_to_iff H)
 
-end decidable
+end decidable

library/standard/logic/classes/inhabited.lean

@@ -5,15 +5,19 @@
 import logic.connectives.basic
 
 inductive inhabited (A : Type) : Type :=
-| inhabited_intro : A → inhabited A
+| inhabited_mk : A → inhabited A
 
-definition inhabited_elim {A : Type} {B : Type} (H1 : inhabited A) (H2 : A → B) : B :=
+namespace inhabited
+
+definition inhabited_destruct {A : Type} {B : Type} (H1 : inhabited A) (H2 : A → B) : B :=
 inhabited_rec H2 H1
 
 definition inhabited_Prop [instance] : inhabited Prop :=
-inhabited_intro true
+inhabited_mk true
 
 definition inhabited_fun [instance] (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B) :=
-inhabited_elim H (take b, inhabited_intro (λa, b))
+inhabited_destruct H (take b, inhabited_mk (λa, b))
 
-definition default (A : Type) {H : inhabited A} : A := inhabited_elim H (take a, a)
+definition default (A : Type) {H : inhabited A} : A := inhabited_destruct H (take a, a)
+
+end inhabited

library/standard/logic/classes/nonempty.lean

@@ -4,6 +4,10 @@
 
 import logic.connectives.basic .inhabited
 
+using inhabited
+
+namespace nonempty
+
 inductive nonempty (A : Type) : Prop :=
 | nonempty_intro : A → nonempty A
 
@@ -12,3 +16,5 @@ nonempty_rec H2 H1
 
 theorem inhabited_imp_nonempty [instance] {A : Type} (H : inhabited A) : nonempty A :=
 nonempty_intro (default A)
+
+end nonempty

library/standard/logic/connectives/basic.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.
 -- Authors: Leonardo de Moura, Jeremy Avigad
-----------------------------------------------------------------------------------------------------
 
 import general_notation .prop
 
@@ -57,6 +55,9 @@ assume Hna : ¬a, absurd (assume Ha : a, absurd_elim b Ha Hna) H
 theorem not_implies_right {a b : Prop} (H : ¬(a → b)) : ¬b :=
 assume Hb : b, absurd (assume Ha : a, Hb) H
 
+theorem contrapos {a b : Prop} (Hab : a → b) : (¬b → ¬a) :=
+assume Hnb Ha, Hnb (Hab Ha)
+
 
 -- and
 -- ---

library/standard/logic/connectives/cast.lean

@@ -6,7 +6,7 @@
 
 import .eq .quantifiers
 
-using eq_proofs
+using eq_ops
 
 definition cast {A B : Type} (H : A = B) (a : A) : B :=
 eq_rec a H
@@ -91,13 +91,13 @@ heq_to_eq (calc cast Hbc (cast Hab a)   == cast Hab a        : cast_heq Hbc (cas
                                    ...  == a                 : cast_heq Hab a
                                    ...  == cast (Hab ⬝ Hbc) a : hsymm (cast_heq (Hab ⬝ Hbc) a))
 
-theorem dcongr2 {A : Type} {B : A → Type} (f : Πx, B x) {a b : A} (H : a = b) : f a == f b :=
+theorem dcongr_arg {A : Type} {B : A → Type} (f : Πx, B x) {a b : A} (H : a = b) : f a == f b :=
 have e1 : ∀ (H : B a = B a), cast H (f a) = f a, from
   assume H, cast_eq H (f a),
 have e2 : ∀ (H : B a = B b), cast H (f a) = f b, from
   subst H e1,
-have e3 : cast (congr2 B H) (f a) = f b, from
+have e3 : cast (congr_arg B H) (f a) = f b, from
-  e2 (congr2 B H),
+  e2 (congr_arg B H),
 cast_eq_to_heq e3
 
 theorem pi_eq {A : Type} {B B' : A → Type} (H : B = B') : (Π x, B x) = (Π x, B' x) :=
@@ -106,20 +106,20 @@ subst H (refl (Π x, B x))
 theorem cast_app' {A : Type} {B B' : A → Type} (H : B = B') (f : Π x, B x) (a : A) :
   cast (pi_eq H) f a == f a :=
 have H1 : ∀ (H : (Π x, B x) = (Π x, B x)), cast H f a == f a, from
-  assume H, eq_to_heq (congr1 (cast_eq H f) a),
+  assume H, eq_to_heq (congr_fun (cast_eq H f) a),
 have H2 : ∀ (H : (Π x, B x) = (Π x, B' x)), cast H f a == f a, from
   subst H H1,
 H2 (pi_eq H)
 
 theorem cast_pull {A : Type} {B B' : A → Type} (H : B = B') (f : Π x, B x) (a : A) :
-                  cast (pi_eq H) f a = cast (congr1 H a) (f a) :=
+                  cast (pi_eq H) f a = cast (congr_fun H a) (f a) :=
 heq_to_eq (calc cast (pi_eq H) f a == f a                     : cast_app' H f a
-                             ...   == cast (congr1 H a) (f a) : hsymm (cast_heq (congr1 H a) (f a)))
+                             ...   == cast (congr_fun H a) (f a) : hsymm (cast_heq (congr_fun H a) (f a)))
 
-theorem hcongr1' {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
+theorem hcongr_fun' {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
     (H1 : f == f') (H2 : B = B')
   : f a == f' a :=
 heq_elim H1 (λ (Ht : (Π x, B x) = (Π x, B' x)) (Hw : cast Ht f = f'),
   calc f a == cast (pi_eq H2) f a  : hsymm (cast_app' H2 f a)
        ... =  cast Ht f a          : refl (cast Ht f a)
-       ... =  f' a                 : congr1 Hw a)
+       ... =  f' a                 : congr_fun Hw a)

library/standard/logic/connectives/eq.lean

@@ -54,27 +54,27 @@ theorem eq_rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (
   from eq_rec_on H2 (take (H2 : b = b), eq_rec_on_id H2 _))
 H2
 
-namespace eq_proofs
+namespace eq_ops
   postfix `⁻¹`:100 := symm
   infixr `⬝`:75     := trans
   infixr `▸`:75    := subst
-end eq_proofs
+end eq_ops
-using eq_proofs
+using eq_ops
 
-theorem congr1 {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
+theorem congr_fun {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
 H ▸ refl (f a)
 
-theorem congr2 {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
+theorem congr_arg {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
 H ▸ refl (f a)
 
 theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) : f a = g b :=
 H1 ▸ H2 ▸ refl (f a)
 
 theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
-take x, congr1 H x
+take x, congr_fun H x
 
 theorem not_congr {a b : Prop} (H : a = b) : (¬a) = (¬b) :=
-congr2 not H
+congr_arg not H
 
 theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
 H1 ▸ H2

library/standard/logic/connectives/examples/instances_test.lean

@@ -6,11 +6,12 @@ import ..instances
 
 using relation
 
+using relation.general_operations
+using relation.iff_ops
+using eq_ops
 
 section
 
-using relation.operations
-
 theorem test1 (a b : Prop) (H : a ↔ b) (H1 : a) : b := mp H H1
 
 end
@@ -18,10 +19,8 @@ end
 
 section
 
-using gensubst
-
 theorem test2 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
-subst H1 H2
+subst iff H1 H2
 
 theorem test3 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
 H1 ▸ H2
@@ -35,12 +34,16 @@ congr.infer iff iff (λa, (a ∨ c → ¬(d → a))) H1
 
 section
 
-using relation.symbols
-
 theorem test5 (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
 H1 ⬝ H2⁻¹ ⬝ H3
 
 theorem test6 (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
 H1 ⬝ (H2⁻¹ ⬝ H3)
 
+theorem test7 (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
+trans H1 (trans (symm H2) H3)
+
+theorem test8 (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
+trans iff H1 (trans iff (symm iff H2) H3)
+
 end

library/standard/logic/connectives/if.lean

@@ -5,7 +5,7 @@
 ----------------------------------------------------------------------------------------------------
 
 import logic.classes.decidable tools.tactic
-using decidable tactic eq_proofs
+using decidable tactic eq_ops
 
 definition ite (c : Prop) {H : decidable c} {A : Type} (t e : A) : A :=
 rec_on H (assume Hc,  t) (assume Hnc, e)

library/standard/logic/connectives/instances.lean

@@ -4,44 +4,46 @@
 
 import logic.connectives.basic logic.connectives.eq struc.relation
 
+namespace relation
+
 using relation
 
 -- Congruences for logic
 -- ---------------------
 
-theorem congr_not : congr.class iff iff not :=
+theorem congr_not : congr iff iff not :=
-congr.mk
+congr_mk
   (take a b,
     assume H : a ↔ b, iff_intro
       (assume H1 : ¬a, assume H2 : b, H1 (iff_elim_right H H2))
       (assume H1 : ¬b, assume H2 : a, H1 (iff_elim_left H H2)))
 
-theorem congr_and : congr.class2 iff iff iff and :=
+theorem congr_and : congr2 iff iff iff and :=
-congr.mk2
+congr2_mk
   (take a1 b1 a2 b2,
     assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
     iff_intro
       (assume H3 : a1 ∧ a2, and_imp_and H3 (iff_elim_left H1) (iff_elim_left H2))
       (assume H3 : b1 ∧ b2, and_imp_and H3 (iff_elim_right H1) (iff_elim_right H2)))
 
-theorem congr_or : congr.class2 iff iff iff or :=
+theorem congr_or : congr2 iff iff iff or :=
-congr.mk2
+congr2_mk
   (take a1 b1 a2 b2,
     assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
     iff_intro
       (assume H3 : a1 ∨ a2, or_imp_or H3 (iff_elim_left H1) (iff_elim_left H2))
       (assume H3 : b1 ∨ b2, or_imp_or H3 (iff_elim_right H1) (iff_elim_right H2)))
 
-theorem congr_imp : congr.class2 iff iff iff imp :=
+theorem congr_imp : congr2 iff iff iff imp :=
-congr.mk2
+congr2_mk
   (take a1 b1 a2 b2,
     assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
     iff_intro
       (assume H3 : a1 → a2, assume Hb1 : b1, iff_elim_left H2 (H3 ((iff_elim_right H1) Hb1)))
       (assume H3 : b1 → b2, assume Ha1 : a1, iff_elim_right H2 (H3 ((iff_elim_left H1) Ha1))))
 
-theorem congr_iff : congr.class2 iff iff iff iff :=
+theorem congr_iff : congr2 iff iff iff iff :=
-congr.mk2
+congr2_mk
   (take a1 b1 a2 b2,
     assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
     iff_intro
@@ -59,62 +61,87 @@ theorem congr_iff_compose [instance] := congr.compose21 congr_iff
 -- Generalized substitution
 -- ------------------------
 
-namespace gensubst
-
 -- TODO: note that the target has to be "iff". Otherwise, there is not enough
 -- information to infer an mp-like relation.
 
-theorem subst {T : Type} {R : T → T → Prop} {P : T → Prop} {C : congr.class R iff P}
+namespace general_operations
+
+theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ {C : congr R iff P}
   {a b : T} (H : R a b) (H1 : P a) : P b := iff_elim_left (congr.app C H) H1
 
-infixr `▸`:75    := subst
+end general_operations
-
-end gensubst
-
 
 -- = is an equivalence relation
 -- ----------------------------
 
-theorem is_reflexive_eq [instance] (T : Type) : relation.is_reflexive.class (@eq T) :=
+theorem is_reflexive_eq [instance] (T : Type) : relation.is_reflexive (@eq T) :=
-relation.is_reflexive.mk (@refl T)
+relation.is_reflexive_mk (@refl T)
 
-theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric.class (@eq T) :=
+theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric (@eq T) :=
-relation.is_symmetric.mk (@symm T)
+relation.is_symmetric_mk (@symm T)
 
-theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive.class (@eq T) :=
+theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive (@eq T) :=
-relation.is_transitive.mk (@trans T)
+relation.is_transitive_mk (@trans T)
+
+-- TODO: this is only temporary, needed to inform Lean that is_equivalence is a class
+theorem is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
+relation.is_equivalence_mk _ _ _
 
 
 -- iff is an equivalence relation
 -- ------------------------------
 
-theorem is_reflexive_iff [instance] : relation.is_reflexive.class iff :=
+theorem is_reflexive_iff [instance] : relation.is_reflexive iff :=
-relation.is_reflexive.mk (@iff_refl)
+relation.is_reflexive_mk (@iff_refl)
 
-theorem is_symmetric_iff [instance] : relation.is_symmetric.class iff :=
+theorem is_symmetric_iff [instance] : relation.is_symmetric iff :=
-relation.is_symmetric.mk (@iff_symm)
+relation.is_symmetric_mk (@iff_symm)
 
-theorem is_transitive_iff [instance] : relation.is_transitive.class iff :=
+theorem is_transitive_iff [instance] : relation.is_transitive iff :=
-relation.is_transitive.mk (@iff_trans)
+relation.is_transitive_mk (@iff_trans)
 
 
 -- Mp-like for iff
 -- ---------------
 
-theorem mp_like_iff [instance] (a b : Prop) (H : a ↔ b) : relation.mp_like.class H :=
+theorem mp_like_iff [instance] (a b : Prop) (H : a ↔ b) : relation.mp_like H :=
-relation.mp_like.mk (iff_elim_left H)
+relation.mp_like_mk (iff_elim_left H)
+
+-- Substition for iff
+-- ------------------
+
+theorem subst_iff {P : Prop → Prop} {C : congr iff iff P} {a b : Prop} (H : a ↔ b) (H1 : P a) :
+  P b :=
+@general_operations.subst Prop iff P C a b H H1
+
+-- Support for calc
+-- ----------------
+
+calc_refl iff_refl
+calc_subst subst_iff
+calc_trans iff_trans
+
+namespace iff_ops
+  postfix `⁻¹`:100 := iff_symm
+  infixr `⬝`:75     := iff_trans
+  infixr `▸`:75    := subst_iff
+end iff_ops
+
+
+
 
 
 -- Boolean calculations
 -- --------------------
 
--- TODO: move these to new file
+-- TODO: move these somewhere
--- TODO: declare trans
 
 theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
 calc
   (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
-    ... ↔ a ∨ (c ∨ b) : congr.infer iff iff _ (or_comm b c)
+    ... ↔ a ∨ (c ∨ b) : {or_comm b c}
-    ... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
+     ... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
 
 -- TODO: add or_left_comm, and_right_comm, and_left_comm
+
+end relation

library/standard/logic/connectives/quantifiers.lean

@@ -4,6 +4,8 @@
 
 import .basic .eq ..classes.nonempty
 
+using inhabited nonempty
+
 inductive Exists {A : Type} (P : A → Prop) : Prop :=
 | exists_intro : ∀ (a : A), P a → Exists P
 
@@ -52,12 +54,12 @@ theorem forall_true_iff_true (A : Type) : (∀x : A, true) ↔ true :=
 iff_intro (assume H, trivial) (assume H, take x, trivial)
 
 theorem forall_p_iff_p (A : Type) {H : inhabited A} (p : Prop) : (∀x : A, p) ↔ p :=
-iff_intro (assume Hl, inhabited_elim H (take x, Hl x)) (assume Hr, take x, Hr)
+iff_intro (assume Hl, inhabited_destruct H (take x, Hl x)) (assume Hr, take x, Hr)
 
 theorem exists_p_iff_p (A : Type) {H : inhabited A} (p : Prop) : (∃x : A, p) ↔ p :=
 iff_intro
   (assume Hl, obtain a Hp, from Hl, Hp)
-  (assume Hr, inhabited_elim H (take a, exists_intro a Hr))
+  (assume Hr, inhabited_destruct H (take a, exists_intro a Hr))
 
 theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) : (∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
 iff_intro

library/standard/struc/binary.lean

@@ -5,7 +5,7 @@
 ----------------------------------------------------------------------------------------------------
 
 import logic.connectives.eq
-using eq_proofs
+using eq_ops
 
 namespace binary
 section

library/standard/struc/equivalence.lean

@@ -1,31 +0,0 @@
-----------------------------------------------------------------------------------------------------
--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Leonardo de Moura
-----------------------------------------------------------------------------------------------------
-
-import logic.connectives.prop
-
-namespace equivalence
-section
-  parameter {A : Type}
-  parameter p : A → A → Prop
-  infix `∼`:50 := p
-  definition reflexive  := ∀a, a ∼ a
-  definition symmetric  := ∀a b, a ∼ b → b ∼ a
-  definition transitive := ∀a b c, a ∼ b → b ∼ c → a ∼ c
-end
-
-inductive equivalence {A : Type} (p : A → A → Prop) : Prop :=
-| equivalence_intro : reflexive p → symmetric p → transitive p → equivalence p
-
-theorem equivalence_reflexive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : reflexive p :=
-equivalence_rec (λ r s t, r) H
-
-theorem equivalence_symmetric [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : symmetric p :=
-equivalence_rec (λ r s t, s) H
-
-theorem equivalence_transitive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : transitive p :=
-equivalence_rec (λ r s t, t) H
-
-end equivalence

library/standard/struc/relation.lean

@@ -4,69 +4,72 @@
 
 import logic.connectives.prop
 
+
 -- General properties of relations
 -- -------------------------------
 
 namespace relation
 
 abbreviation reflexive {T : Type} (R : T → T → Type) : Type := ∀x, R x x
-abbreviation symmetric {T : Type} (R : T → T → Type) : Type := ∀x y, R x y → R y x
+abbreviation symmetric {T : Type} (R : T → T → Type) : Type := ∀⦃x y⦄, R x y → R y x
-abbreviation transitive {T : Type} (R : T → T → Type) : Type := ∀x y z, R x y → R y z → R x z
+abbreviation transitive {T : Type} (R : T → T → Type) : Type := ∀⦃x y z⦄, R x y → R y z → R x z
+
+
+inductive is_reflexive {T : Type} (R : T → T → Type) : Prop :=
+| is_reflexive_mk : reflexive R → is_reflexive R
 
 namespace is_reflexive
 
-  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : is_reflexive R) : reflexive R :=
-  | mk : reflexive R → class R
+  is_reflexive_rec (λu, u) C
 
-  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) : reflexive R :=
+  abbreviation infer ⦃T : Type⦄ (R : T → T → Type) {C : is_reflexive R} : reflexive R :=
-  class_rec (λu, u) C
+  is_reflexive_rec (λu, u) C
-
-  abbreviation infer ⦃T : Type⦄ {R : T → T → Type} {C : class R} : reflexive R :=
-  class_rec (λu, u) C
 
 end is_reflexive
 
+
+inductive is_symmetric {T : Type} (R : T → T → Type) : Prop :=
+| is_symmetric_mk : symmetric R → is_symmetric R
+
 namespace is_symmetric
 
-  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : is_symmetric R) : symmetric R :=
-  | mk : symmetric R → class R
+  is_symmetric_rec (λu, u) C
 
-  abbreviation app ⦃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 : is_symmetric R} : symmetric R :=
-  class_rec (λu, u) C x y H
+  is_symmetric_rec (λu, u) C
-
-  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
 
 end is_symmetric
 
+
+inductive is_transitive {T : Type} (R : T → T → Type) : Prop :=
+| is_transitive_mk : transitive R → is_transitive R
+
 namespace is_transitive
 
-  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : is_transitive R) : transitive R :=
-  | mk : transitive R → class R
+  is_transitive_rec (λu, u) C
 
-  abbreviation app ⦃T : Type⦄ {R : T → T → Type} (C : class R) ⦃x y z : T⦄ (H1 : R x y)
+  abbreviation infer ⦃T : Type⦄ (R : T → T → Type) {C : is_transitive R} : transitive R :=
-    (H2 : R y z) : R x z :=
+  is_transitive_rec (λu, u) C
-  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 :=
-  class_rec (λu, u) C x y z H1 H2
 
 end is_transitive
 
+
+inductive is_equivalence {T : Type} (R : T → T → Type) : Prop :=
+| is_equivalence_mk : is_reflexive R → is_symmetric R → is_transitive R → is_equivalence R
+
 namespace is_equivalence
 
-  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  theorem is_reflexive {T : Type} (R : T → T → Type) {C : is_equivalence R} : is_reflexive R :=
-  | mk : is_reflexive.class R → is_symmetric.class R → is_transitive.class R → class R
+  is_equivalence_rec (λx y z, x) C
 
-  theorem is_reflexive {T : Type} {R : T → T → Type} {C : class R} : is_reflexive.class R :=
+  theorem is_symmetric {T : Type} {R : T → T → Type} {C : is_equivalence R} : is_symmetric R :=
-  class_rec (λx y z, x) C
+  is_equivalence_rec (λx y z, y) C
 
-  theorem is_symmetric {T : Type} {R : T → T → Type} {C : class R} : is_symmetric.class R :=
+  theorem is_transitive {T : Type} {R : T → T → Type} {C : is_equivalence R} : is_transitive R :=
-  class_rec (λx y z, y) C
+  is_equivalence_rec (λx y z, z) 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
 
@@ -74,112 +77,109 @@ instance is_equivalence.is_reflexive
 instance is_equivalence.is_symmetric
 instance is_equivalence.is_transitive
 
+
+-- partial equivalence relation
+inductive is_PER {T : Type} (R : T → T → Type) : Prop :=
+| is_PER_mk : is_symmetric R → is_transitive R → is_PER R
+
 namespace is_PER
 
-  inductive class {T : Type} (R : T → T → Type) : Prop :=
+  theorem is_symmetric {T : Type} {R : T → T → Type} {C : is_PER R} : is_symmetric R :=
-  | mk : is_symmetric.class R → is_transitive.class R → class R
+  is_PER_rec (λx y, x) C
 
-  theorem is_symmetric {T : Type} {R : T → T → Type} {C : class R} : is_symmetric.class R :=
+  theorem is_transitive {T : Type} {R : T → T → Type} {C : is_PER R} : is_transitive R :=
-  class_rec (λx y, x) C
+  is_PER_rec (λx y, y) 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_symmetric
 instance is_PER.is_transitive
 
 
 -- Congruence for unary and binary functions
 -- -----------------------------------------
 
-namespace congr
+inductive congr {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
-
-inductive class {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
     (f : T1 → T2) : Prop :=
-| mk : (∀x y, R1 x y → R2 (f x) (f y)) → class R1 R2 f
+| congr_mk : (∀x y, R1 x y → R2 (f x) (f y)) → congr R1 R2 f
-
-abbreviation app {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
-    {f : T1 → T2} (C : class R1 R2 f) ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
-class_rec (λu, u) C x y
-
-theorem infer {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
-    (f : T1 → T2) {C : class R1 R2 f} ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
-class_rec (λu, u) C x y
 
 -- for binary functions
-inductive class2 {T1 : Type}  (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
+inductive congr2 {T1 : Type}  (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
     {T3 : Type} (R3 : T3 → T3 → Prop) (f : T1 → T2 → T3) : Prop :=
-| mk2 : (∀(x1 y1 : T1) (x2 y2 : T2), R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2)) →
+| congr2_mk : (∀(x1 y1 : T1) (x2 y2 : T2), R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2)) →
-    class2 R1 R2 R3 f
+    congr2 R1 R2 R3 f
 
-abbreviation app2 {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
+namespace congr
-    {T3 : Type} {R3 : T3 → T3 → Prop}
+
-    {f : T1 → T2 → T3} (C : class2 R1 R2 R3 f) ⦃x1 y1 : T1⦄ ⦃x2 y2 : T2⦄
+  abbreviation app {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
-  : R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2) :=
+      {f : T1 → T2} (C : congr R1 R2 f) ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
-class2_rec (λu, u) C x1 y1 x2 y2
+  congr_rec (λu, u) C x y
+
+  theorem infer {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
+      (f : T1 → T2) {C : congr R1 R2 f} ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
+  congr_rec (λu, u) C x y
+
+  abbreviation app2 {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
+      {T3 : Type} {R3 : T3 → T3 → Prop}
+      {f : T1 → T2 → T3} (C : congr2 R1 R2 R3 f) ⦃x1 y1 : T1⦄ ⦃x2 y2 : T2⦄ :
+    R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2) :=
+  congr2_rec (λu, u) C x1 y1 x2 y2
 
 -- ### general tools to build instances
 
-theorem compose
+  theorem compose
-    {T2 : Type} {R2 : T2 → T2 → Prop}
+      {T2 : Type} {R2 : T2 → T2 → Prop}
-    {T3 : Type} {R3 : T3 → T3 → Prop}
+      {T3 : Type} {R3 : T3 → T3 → Prop}
-    {g : T2 → T3} (C2 : congr.class R2 R3 g)
+      {g : T2 → T3} (C2 : congr R2 R3 g)
-    {{T1 : Type}} {R1 : T1 → T1 → Prop}
+      ⦃T1 : Type⦄ {R1 : T1 → T1 → Prop}
-    {f : T1 → T2} (C1 : congr.class R1 R2 f) :
+      {f : T1 → T2} (C1 : congr R1 R2 f) :
-  congr.class R1 R3 (λx, g (f x)) :=
+    congr R1 R3 (λx, g (f x)) :=
-mk (λx1 x2 H, app C2 (app C1 H))
+  congr_mk (λx1 x2 H, app C2 (app C1 H))
 
-theorem compose21
+  theorem compose21
-    {T2 : Type} {R2 : T2 → T2 → Prop}
+      {T2 : Type} {R2 : T2 → T2 → Prop}
-    {T3 : Type} {R3 : T3 → T3 → Prop}
+      {T3 : Type} {R3 : T3 → T3 → Prop}
-    {T4 : Type} {R4 : T4 → T4 → Prop}
+      {T4 : Type} {R4 : T4 → T4 → Prop}
-    {g : T2 → T3 → T4} (C3 : congr.class2 R2 R3 R4 g)
+      {g : T2 → T3 → T4} (C3 : congr2 R2 R3 R4 g)
-    ⦃T1 : Type⦄ {R1 : T1 → T1 → Prop}
+      ⦃T1 : Type⦄ {R1 : T1 → T1 → Prop}
-    {f1 : T1 → T2} (C1 : congr.class R1 R2 f1)
+      {f1 : T1 → T2} (C1 : congr R1 R2 f1)
-    {f2 : T1 → T3} (C2 : congr.class R1 R3 f2) :
+      {f2 : T1 → T3} (C2 : congr R1 R3 f2) :
-  congr.class R1 R4 (λx, g (f1 x) (f2 x)) :=
+    congr R1 R4 (λx, g (f1 x) (f2 x)) :=
-mk (λx1 x2 H, app2 C3 (app C1 H) (app C2 H))
+  congr_mk (λx1 x2 H, app2 C3 (app C1 H) (app C2 H))
 
-theorem const {T2 : Type} (R2 : T2 → T2 → Prop) (H : relation.reflexive R2)
+  theorem const {T2 : Type} (R2 : T2 → T2 → Prop) (H : relation.reflexive R2)
-    ⦃T1 : Type⦄ (R1 : T1 → T1 → Prop) (c : T2) :
+      ⦃T1 : Type⦄ (R1 : T1 → T1 → Prop) (c : T2) :
-  class R1 R2 (λu : T1, c) :=
+    congr R1 R2 (λu : T1, c) :=
-mk (λx y H1, H c)
+  congr_mk (λx y H1, H c)
 
 end congr
 
-end relation
+-- Notice these can't be in the congr namespace, if we want it visible without
-
-
--- TODO: notice these can't be in the congr namespace, if we want it visible without
 -- using congr.
 
 theorem congr_const [instance] {T2 : Type} (R2 : T2 → T2 → Prop)
-    {C : relation.is_reflexive.class R2} ⦃T1 : Type⦄ (R1 : T1 → T1 → Prop) (c : T2) :
+    {C : is_reflexive R2} ⦃T1 : Type⦄ (R1 : T1 → T1 → Prop) (c : T2) :
-  relation.congr.class R1 R2 (λu : T1, c) :=
+  congr R1 R2 (λu : T1, c) :=
-relation.congr.const R2 (relation.is_reflexive.app C) R1 c
+congr.const R2 (is_reflexive.app C) R1 c
 
 theorem congr_trivial [instance] {T : Type} (R : T → T → Prop) :
-  relation.congr.class R R (λu, u) :=
+  congr R R (λu, u) :=
-relation.congr.mk (λx y H, H)
+congr_mk (λx y H, H)
 
 
 -- Relations that can be coerced to functions / implications
 -- ---------------------------------------------------------
 
-namespace relation
+inductive mp_like {R : Type → Type → Prop} {a b : Type} (H : R a b) : Prop :=
+| mp_like_mk {} : (a → b) → @mp_like R a b H
 
 namespace mp_like
 
-inductive class {R : Type → Type → Prop} {a b : Type} (H : R a b) : Prop :=
+  definition app {R : Type → Type → Prop} {a : Type} {b : Type} {H : R a b}
-| mk {} : (a → b) → @class R a b H
+    (C : mp_like H) : a → b := mp_like_rec (λx, x) C
 
-definition app {R : Type → Type → Prop} {a : Type} {b : Type} {H : R a b}
+  definition infer ⦃R : Type → Type → Prop⦄ {a : Type} {b : Type} (H : R a b)
-  (C : class H) : a → b := class_rec (λx, x) C
+    {C : mp_like H} : a → b := mp_like_rec (λx, x) C
-
-definition infer ⦃R : Type → Type → Prop⦄ {a : Type} {b : Type} (H : R a b)
-  {C : class H} : a → b := class_rec (λx, x) C
 
 end mp_like
 
@@ -187,20 +187,21 @@ end mp_like
 -- Notation for operations on general symbols
 -- ------------------------------------------
 
-namespace operations
+namespace general_operations
 
+-- e.g. if R is an instance of the class, then "refl R" is reflexivity for the class
 definition refl := is_reflexive.infer
 definition symm := is_symmetric.infer
 definition trans := is_transitive.infer
 definition mp := mp_like.infer
 
-end operations
+end general_operations
 
-namespace symbols
+-- namespace
+--
+-- postfix `⁻¹`:100 := operations.symm
+-- infixr `⬝`:75     := operations.trans
 
-postfix `⁻¹`:100 := operations.symm
+-- end symbols
-infixr `⬝`:75     := operations.trans
-
-end symbols
 
 end relation

library/standard/tools/fake_simplifier.lean

@@ -0,0 +1,9 @@
+import .tactic
+using tactic
+
+namespace fake_simplifier
+
+-- until we have the simplifier...
+definition simp : tactic := apply @sorry
+
+end fake_simplifier

library/standard/tools/tactic.lean

@@ -54,4 +54,5 @@ notation `?` t:max     := try t
 definition repeat1     (t : tactic) : tactic := t ; !t
 definition focus       (t : tactic) : tactic := focus_at t 0
 definition determ      (t : tactic) : tactic := at_most t 1
+
 end tactic