feat(types.nat): prove that inequalities on nat are mere propositions

Also some small changes in various other locations

hott/hit/interval.hlean

@@ -1,7 +1,6 @@
 /-
 Copyright (c) 2015 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-
 Authors: Floris van Doorn
 
 Declaration of the interval

hott/init/funext.hlean

@@ -256,3 +256,6 @@ definition naive_funext_of_ua : naive_funext :=
 protected definition homotopy.rec_on [recursor] {Q : (f ∼ g) → Type} (p : f ∼ g)
   (H : Π(q : f = g), Q (apd10 q)) : Q p :=
 right_inv apd10 p ▸ H (eq_of_homotopy p)
+
+protected definition homotopy.rec_on_idp [recursor] {Q : Π{g}, (f ∼ g) → Type} {g : Π x, P x} (p : f ∼ g) (H : Q (homotopy.refl f)) : Q p :=
+homotopy.rec_on p (λq, eq.rec_on q H)

hott/init/pathover.hlean

@@ -1,7 +1,6 @@
 /-
 Copyright (c) 2015 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-
 Author: Floris van Doorn
 
 Basic theorems about pathovers

hott/init/trunc.hlean

@@ -38,13 +38,12 @@ namespace is_trunc
 
   definition leq (n m : trunc_index) : Type₀ :=
   trunc_index.rec_on n (λm, unit) (λ n p m, trunc_index.rec_on m (λ p, empty) (λ m q p, p m) p) m
+  notation x <= y := trunc_index.leq x y
+  notation x ≤ y := trunc_index.leq x y
   end trunc_index
 
   infix `+2+`:65 := trunc_index.add
 
-  notation x <= y := trunc_index.leq x y
-  notation x ≤ y := trunc_index.leq x y
-
   namespace trunc_index
   definition succ_le_succ {n m : trunc_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
   definition le_of_succ_le_succ {n m : trunc_index} (H : n.+1 ≤ m.+1) : n ≤ m := H

hott/types/nat/default.hlean

@@ -4,4 +4,4 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
 
-import .basic .order .sub
+import .basic .order .sub .hott

hott/types/nat/hott.hlean

@@ -0,0 +1,32 @@
+/-
+Copyright (c) 2015 Floris van Doorn. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Floris van Doorn
+
+Theorems about the natural numbers specific to HoTT
+-/
+
+import .basic
+
+open is_trunc unit empty eq
+
+namespace nat
+
+  definition is_hprop_lt [instance] (n m : ℕ) : is_hprop (n < m) :=
+  begin
+    assert H : Π{n m : ℕ} (p : n < m) (q : succ n = m), p = q ▸ lt.base n,
+    { intros, cases p,
+      { assert H' : q = idp, apply is_hset.elim,
+        cases H', reflexivity},
+      { cases q, exfalso, exact lt.irrefl b a}},
+    apply is_hprop.mk, intros p q,
+    induction q,
+    { apply H},
+    { cases p,
+        exfalso, exact lt.irrefl b a,
+        exact ap lt.step !v_0}
+  end
+
+  definition is_hprop_le (n m : ℕ) : is_hprop (n ≤ m) := !is_hprop_lt
+
+end nat

hott/types/nat/order.hlean

@@ -9,7 +9,7 @@ Ported from standard library
 -/
 
 import .basic algebra.ordered_ring
-open core prod decidable
+open prod decidable sum eq
 
 namespace nat
 

hott/types/squareover.hlean

@@ -1,7 +1,6 @@
 /-
 Copyright (c) 2015 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-
 Author: Floris van Doorn
 
 Theorems about squareovers

hott/types/trunc.hlean

@@ -10,7 +10,7 @@ Properties of is_trunc and trunctype
 
 import types.pi types.eq types.equiv .function
 
-open eq sigma sigma.ops pi function equiv is_trunc.trunctype is_equiv prod
+open eq sigma sigma.ops pi function equiv is_trunc.trunctype is_equiv prod is_trunc.trunc_index
 
 namespace is_trunc
   variables {A B : Type} {n : trunc_index}

src/emacs/lean-syntax.el

@@ -135,7 +135,7 @@
      (,(rx (not (any "\.")) word-start
            (group
             (or "\\b.*_tac" "Cond" "or_else" "then" "try" "when" "assumption" "eassumption" "rapply"
-                "apply" "fapply" "eapply" "rename" "intro" "intros" "all_goals" "fold"
+                "apply" "fapply" "eapply" "rename" "intro" "intros" "all_goals" "fold" "focus" "focus_at"
                 "generalize" "generalizes" "clear" "clears" "revert" "reverts" "back" "beta" "done" "exact" "rexact"
                 "refine" "repeat" "whnf" "rotate" "rotate_left" "rotate_right" "inversion" "cases" "rewrite" "esimp"
                 "unfold" "change" "check_expr" "contradiction" "exfalso" "split" "existsi" "constructor" "left" "right"