feat(nat): redefine le and lt in the standard library

hott/init/nat.hlean

@@ -30,6 +30,7 @@ namespace nat
   nat.cases_on a zero (λ a₁, a₁)
 
   -- add is defined in init.num
+
   definition sub (a b : nat) : nat :=
   nat.rec_on b a (λ b₁ r, pred r)
 
@@ -57,49 +58,52 @@ namespace nat
 
   /- properties of inequality -/
 
-  definition le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
+  theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
 
-  definition le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
+  theorem le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
 
-  definition pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
+  theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
 
-  definition le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
+  theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
   by induction H2 with n H2 IH;exact H1;exact le.step IH
 
-  definition le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
+  theorem le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
 
-  definition le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
+  theorem le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
 
-  definition le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
+  theorem le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
 
-  definition succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
+  theorem succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
   by induction H;reflexivity;exact le.step v_0
 
-  definition pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
+  theorem pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
   by induction H;reflexivity;cases b;exact v_0;exact le.step v_0
 
-  definition le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
+  theorem le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
   pred_le_pred H
 
-  definition le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
+  theorem le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
   by cases n;exact le.step H;exact succ_le_succ H
 
-  definition not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
+  theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
   by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
 
-  definition zero_le (n : ℕ) : 0 ≤ n :=
+  theorem zero_le (n : ℕ) : 0 ≤ n :=
   by induction n with n IH;apply le.refl;exact le.step IH
 
-  definition zero_lt_succ (n : ℕ) : 0 < succ n :=
+  theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
+  le.step H
+
+  theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
   by induction n with n IH;apply le.refl;exact le.step IH
 
-  definition lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
+  theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
   le.trans (le.step H1) H2
 
-  definition lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
+  theorem lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
   le.trans (succ_le_succ H1) H2
 
-  definition lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
+  theorem lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
   le.trans H1 H2
 
   theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
@@ -111,7 +115,7 @@ namespace nat
       { exfalso, apply not_succ_le_self, exact lt.trans H1' H2'}},
   end
 
-  definition not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
+  theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
   by intro H; cases H
 
   theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
@@ -128,7 +132,7 @@ namespace nat
   theorem lt.asymm {n m : ℕ} (H1 : n < m) (H2 : m < n) : empty :=
   le_lt_antisymm (le_of_lt H1) H2
 
-  theorem lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
+  definition lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
   begin
     revert b H1 H2 H3, induction a with a IH,
     { intros, cases b,
@@ -142,16 +146,16 @@ namespace nat
           intro H, exact H3 (succ_le_succ H)}}
   end
 
-  definition lt.trichotomy (a b : ℕ) : a < b ⊎ a = b ⊎ b < a :=
+  theorem lt.trichotomy (a b : ℕ) : a < b ⊎ a = b ⊎ b < a :=
-  lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
+  lt.by_cases inl (λH, inr (inl H)) (λH, inr (inr H))
-
-  theorem lt_or_ge (a b : ℕ) : (a < b) ⊎ (a ≥ b) :=
-  lt.by_cases inl (λH, inr (eq.rec_on H !le.refl)) (λH, inr (le_of_lt H))
 
   definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
-  sum.rec_on (lt_or_ge a b) H1 H2
+  lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
 
-  definition not_lt_zero (a : ℕ) : ¬ a < zero :=
+  theorem lt_or_ge (a b : ℕ) : (a < b) ⊎ (a ≥ b) :=
+  lt_ge_by_cases inl inr
+
+  theorem not_lt_zero (a : ℕ) : ¬ a < zero :=
   by intro H; cases H
 
   -- less-than is well-founded
@@ -198,49 +202,49 @@ namespace nat
   definition decidable_gt [instance] : decidable_rel gt := _
   definition decidable_ge [instance] : decidable_rel ge := _
 
-  definition eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ⊎ a < b :=
+  theorem eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ⊎ a < b :=
   by cases H with b' H; exact sum.inl rfl; exact sum.inr (succ_le_succ H)
 
 
-  definition le_of_eq_or_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
+  theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
   by cases H with H H; exact le_of_eq H; exact le_of_lt H
 
-  definition eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
+  theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
   sum.rec_on (lt.trichotomy a b)
     (λ hlt, absurd hlt hnlt)
     (λ h, h)
 
-  definition lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
+  theorem lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
   succ_le_succ h
 
-  definition lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
+  theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
 
-  definition succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
+  theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
 
   definition max (a b : ℕ) : ℕ := if a < b then b else a
   definition min (a b : ℕ) : ℕ := if a < b then a else b
 
-  definition max_self (a : ℕ) : max a a = a :=
+  theorem max_self (a : ℕ) : max a a = a :=
   eq.rec_on !if_t_t rfl
 
-  definition max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
+  theorem max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
   if_pos H
 
-  definition max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
+  theorem max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
   if_neg H
 
-  definition eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
+  theorem eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
   eq.rec_on (max_eq_right H) rfl
 
-  definition eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
+  theorem eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
   eq.rec_on (max_eq_left H) rfl
 
-  definition le_max_left (a b : ℕ) : a ≤ max a b :=
+  theorem le_max_left (a b : ℕ) : a ≤ max a b :=
   by_cases
     (λ h : a < b,   le_of_lt (eq.rec_on (eq_max_right h) h))
     (λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
 
-  definition le_max_right (a b : ℕ) : b ≤ max a b :=
+  theorem le_max_right (a b : ℕ) : b ≤ max a b :=
   by_cases
     (λ h : a < b,   eq.rec_on (eq_max_right h) !le.refl)
     (λ h : ¬ a < b, sum.rec_on (eq_or_lt_of_not_lt h)
@@ -249,21 +253,21 @@ namespace nat
         have aux : a = max a b, from eq_max_left (lt.asymm h),
         eq.rec_on aux (le_of_lt h)))
 
-  definition succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
+  theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
   by induction b with b IH; reflexivity; apply ap pred IH
 
-  definition sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
+  theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
   eq.rec_on (succ_sub_succ_eq_sub a b) rfl
 
-  definition zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
+  theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
   nat.rec_on a
     rfl
     (λ a₁ (ih : zero - a₁ = zero), ap pred ih)
 
-  definition zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
+  theorem zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
   eq.rec_on (zero_sub_eq_zero a) rfl
 
-  definition sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
+  theorem sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
   have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
     λa h₁, le.rec_on h₁
       (λb h₂, le.cases_on h₂
@@ -278,7 +282,7 @@ namespace nat
             (lt.trans (@ih b₁ bpos) (lt.base a₁)))),
   λ h₁ h₂, aux h₁ h₂
 
-  definition sub_le (a b : ℕ) : a - b ≤ a :=
+  theorem sub_le (a b : ℕ) : a - b ≤ a :=
   nat.rec_on b
     (le.refl a)
     (λ b₁ ih, le.trans !pred_le ih)

library/data/encodable.lean

@@ -42,7 +42,7 @@ encodable.mk
     begin
     cases o with a,
       begin esimp end,
-      begin esimp, rewrite [if_neg !succ_ne_zero, pred_succ, encodable.encodek] end
+      begin esimp, rewrite [if_neg !succ_ne_zero, encodable.encodek] end
     end)
 
 section sum

library/data/list/set.lean

@@ -462,9 +462,12 @@ assume i, mem_cons_of_mem _ i
 
 theorem mem_upto_of_lt : ∀ {n i : nat}, i < n → i ∈ upto n
 | 0        i h := absurd h !not_lt_zero
-| (succ n) i h := or.elim (eq_or_lt_of_le h)
+| (succ n) i h :=
-  (λ ieqn : i = n, by rewrite [ieqn, upto_succ]; exact !mem_cons)
+begin
-  (λ iltn : i < n, mem_upto_succ_of_mem_upto (mem_upto_of_lt iltn))
+  cases h with m h',
+  { rewrite upto_succ, apply mem_cons},
+  { exact mem_upto_succ_of_mem_upto (mem_upto_of_lt h')}
+end
 
 /- union -/
 section union

library/data/nat/bquant.lean

@@ -36,7 +36,7 @@ namespace nat
 
   definition not_bex_succ {P : nat → Prop} {n : nat} (H₁ : ¬ bex n P) (H₂ : ¬ P n) : ¬ bex (succ n) P :=
   λ H, obtain (w : nat) (Hw : w < succ n ∧ P w), from H,
-    and.rec_on Hw (λ hltsn hp, or.rec_on (eq_or_lt_of_le hltsn)
+    and.rec_on Hw (λ hltsn hp, or.rec_on (eq_or_lt_of_le (le_of_succ_le_succ hltsn))
       (λ heq : w = n, absurd (eq.rec_on heq hp) H₂)
       (λ hltn : w < n, absurd (exists.intro w (and.intro hltn hp)) H₁))
 
@@ -47,7 +47,7 @@ namespace nat
   λ x Hlt, H x (lt.step Hlt)
 
   definition ball_succ_of_ball {n : nat} {P : nat → Prop} (H₁ : ball n P) (H₂ : P n) : ball (succ n) P :=
-  λ (x : nat) (Hlt : x < succ n), or.elim (eq_or_lt_of_le Hlt)
+  λ (x : nat) (Hlt : x < succ n), or.elim (eq_or_lt_of_le (le_of_succ_le_succ Hlt))
     (λ heq : x = n, eq.rec_on (eq.rec_on heq rfl) H₂)
     (λ hlt : x < n, H₁ x hlt)
 

library/data/nat/choose.lean

@@ -24,7 +24,7 @@ private lemma lbp_zero : lbp 0 :=
 
 private lemma lbp_succ {x : nat} : lbp x → ¬ p x → lbp (succ x) :=
 λ lx npx y yltsx,
-  or.elim (eq_or_lt_of_le yltsx)
+  or.elim (eq_or_lt_of_le (le_of_succ_le_succ yltsx))
     (λ yeqx, by substvars; assumption)
     (λ yltx, lx y yltx)
 
@@ -46,7 +46,7 @@ acc.intro x (λ (y : nat) (l : y ≺ x),
    by_cases
      (λ yeqx : y = succ x, by substvars; assumption)
      (λ ynex : y ≠ succ x,
-        have ygtsx : succ x < y, from lt_of_le_and_ne (succ_lt_succ ygtx) (ne.symm ynex),
+        have ygtsx : succ x < y, from lt_of_le_and_ne ygtx (ne.symm ynex),
         acc.inv h (and.intro ygtsx (and.elim_right l))))
 
 private lemma acc_of_px_of_gt {x y : nat} : p x → y > x → acc gtb y :=
@@ -61,7 +61,7 @@ private lemma acc_of_acc_of_lt : ∀ {x y : nat}, acc gtb x → y < x → acc gt
   assert ax : acc gtb x, from acc_of_acc_succ asx,
   by_cases
      (λ yeqx : y = x, by substvars; assumption)
-     (λ ynex : y ≠ x, acc_of_acc_of_lt ax (lt_of_le_and_ne yltsx ynex))
+     (λ ynex : y ≠ x, acc_of_acc_of_lt ax (lt_of_le_and_ne (le_of_lt_succ yltsx) ynex))
 
 parameter (ex : ∃ a, p a)
 parameter [dp : decidable_pred p]

library/data/nat/div.lean

@@ -442,7 +442,7 @@ have H3 : m * k < (succ (n div k)) * k, from
       ... = n div k * k + n mod k      : eq_div_mul_add_mod
       ... < n div k * k + k            : add_lt_add_left (!mod_lt H1)
       ... = (succ (n div k)) * k       : succ_mul,
-lt_of_mul_lt_mul_right H3
+le_of_lt_succ (lt_of_mul_lt_mul_right H3)
 
 theorem le_div_iff_mul_le {m n k : ℕ} (H : k > 0) : m ≤ n div k ↔ m * k ≤ n :=
 iff.intro !mul_le_of_le_div (!le_div_of_mul_le H)

library/data/nat/order.lean

@@ -15,12 +15,6 @@ namespace nat
 theorem le_of_lt_or_eq {m n : ℕ} (H : m < n ∨ m = n) : m ≤ n :=
 or.elim H (take H1, le_of_lt H1) (take H1, H1 ▸ !le.refl)
 
-theorem lt.by_cases {a b : ℕ} {P : Prop}
-  (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
-or.elim !lt.trichotomy
-  (assume H, H1 H)
-  (assume H, or.elim H (assume H', H2 H') (assume H', H3 H'))
-
 theorem lt_or_eq_of_le {m n : ℕ} (H : m ≤ n) : m < n ∨ m = n :=
 lt.by_cases
   (assume H1 : m < n, or.inl H1)
@@ -55,15 +49,8 @@ theorem le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
 h ▸ le_add_right n k
 
 theorem le.elim {n m : ℕ} (h : n ≤ m) : ∃k, n + k = m :=
-le.rec_on h
+by induction h with m h ih;existsi 0; reflexivity;
-  (exists.intro 0 rfl)
+     cases ih with k H; existsi succ k; exact congr_arg succ H
-  (λ m (h : n < m), lt.rec_on h
-    (exists.intro 1 rfl)
-    (λ b hlt (ih : ∃ (k : ℕ), n + k = b),
-      obtain (k : ℕ) (h : n + k = b), from ih,
-      exists.intro (succ k) (calc
-        n + succ k = succ (n + k) : add_succ
-               ... = succ b       : h)))
 
 theorem le.total {m n : ℕ} : m ≤ n ∨ n ≤ m :=
 lt.by_cases
@@ -124,25 +111,6 @@ lt_of_lt_of_le H2 H3
 theorem mul_lt_mul_of_pos_right {n m k : ℕ} (H : n < m) (Hk : k > 0) : n * k < m * k :=
 !mul.comm ▸ !mul.comm ▸ mul_lt_mul_of_pos_left H Hk
 
-theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
-obtain (k : ℕ) (Hk : n + k = m), from le.elim H1,
-obtain (l : ℕ) (Hl : m + l = n), from le.elim H2,
-have L1 : k + l = 0, from
-  add.cancel_left
-    (calc
-      n + (k + l) = n + k + l : add.assoc
-        ... = m + l           : by subst m
-        ... = n               : by subst n
-        ... = n + 0           : add_zero),
-assert L2 : k = 0, from eq_zero_of_add_eq_zero_right L1,
-calc
-  n = n + 0  : add_zero
-... = n + k  : by subst k
-... = m      : Hk
-
-theorem zero_le (n : ℕ) : 0 ≤ n :=
-le.intro !zero_add
-
 /- nat is an instance of a linearly ordered semiring -/
 
 section migrate_algebra
@@ -237,17 +205,6 @@ eq_zero_of_add_eq_zero_right Hk
 theorem lt_iff_succ_le (m n : nat) : m < n ↔ succ m ≤ n :=
 iff.intro succ_le_of_lt lt_of_succ_le
 
-theorem not_succ_le_zero (n : ℕ) : ¬ succ n ≤ 0 :=
-(assume H : succ n ≤ 0,
-  have H2 : succ n = 0, from eq_zero_of_le_zero H,
-  absurd H2 !succ_ne_zero)
-
-theorem succ_le_succ {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
-!add_one ▸ !add_one ▸ add_le_add_right H 1
-
-theorem le_of_succ_le_succ {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
-le_of_add_le_add_right (by rewrite -add_one at H; assumption)
-
 theorem self_le_succ (n : ℕ) : n ≤ succ n :=
 le.intro !add_one
 
@@ -256,14 +213,6 @@ or.elim (lt_or_eq_of_le H)
   (assume H1 : n < m, or.inl (succ_le_of_lt H1))
   (assume H1 : n = m, or.inr H1)
 
-theorem le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
-nat.cases_on n
-  (assume H : pred 0 ≤ m, !zero_le)
-  (take n',
-    assume H : pred (succ n') ≤ m,
-    have H1 : n' ≤ m, from pred_succ n' ▸ H,
-    succ_le_succ H1)
-
 theorem pred_le_of_le_succ {n m : ℕ} : n ≤ succ m → pred n ≤ m :=
 nat.cases_on n
   (assume H, !pred_zero⁻¹ ▸ zero_le m)

library/data/nat/sqrt.lean

@@ -34,7 +34,7 @@ theorem sqrt_aux_lower : ∀ {s n : nat}, s ≤ n → sqrt_aux s n * sqrt_aux s
 | (succ s) n h := by_cases
   (λ h₁ : (succ s)*(succ s) ≤ n,   by rewrite [sqrt_aux_succ_of_pos h₁]; exact h₁)
   (λ h₂ : ¬ (succ s)*(succ s) ≤ n,
-     assert aux : s ≤ n, from lt.step (lt_of_succ_le h),
+     assert aux : s ≤ n, from le_of_succ_le h,
      by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_lower aux))
 
 theorem sqrt_lower (n : nat) : sqrt n * sqrt n ≤ n :=
@@ -47,7 +47,7 @@ theorem sqrt_aux_upper : ∀ {s n : nat}, n ≤ s*s + s + s → n ≤ sqrt_aux s
     by rewrite [sqrt_aux_succ_of_pos h₁]; exact h)
   (λ h₂ : ¬ (succ s)*(succ s) ≤ n,
     assert h₃ : n < (succ s) * (succ s), from lt_of_not_ge h₂,
-    assert h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact h₃,
+    assert h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact le_of_lt_succ h₃,
     by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_upper h₄))
 
 theorem sqrt_upper (n : nat) : n ≤ sqrt n * sqrt n + sqrt n + sqrt n :=
@@ -68,7 +68,7 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
 | (succ s) h₂ := by_cases
   (λ hl : (succ s)*(succ s) ≤ n*n + k,
      have   l₁ : n*n + k ≤ n*n + n + n,       from by rewrite [add.assoc]; exact (add_le_add_left h₁ (n*n)),
-     assert l₂ : n*n + k < n*n + n + n + 1,   from l₁,
+     assert l₂ : n*n + k < n*n + n + n + 1,   from lt_succ_of_le l₁,
      have   l₃ : n*n + k < (succ n)*(succ n), by rewrite [-succ_mul_succ_eq at l₂]; exact l₂,
      assert l₄ : (succ s)*(succ s) < (succ n)*(succ n), from lt_of_le_of_lt hl l₃,
      have   ng : ¬ succ s > (succ n), from
@@ -79,7 +79,7 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
      have ssnesn  : succ s ≠ succ n, from
        assume sseqsn : succ s = succ n,
          by rewrite [sseqsn at l₄]; exact (absurd l₄ !lt.irrefl),
-     have   sslen : succ s ≤ n, from lt_of_le_and_ne sslesn ssnesn,
+     have   sslen : s < n, from lt_of_succ_lt_succ (lt_of_le_and_ne sslesn ssnesn),
      assert sseqn : succ s = n, from le.antisymm sslen h₂,
      by rewrite [sqrt_aux_succ_of_pos hl]; exact sseqn)
   (λ hg : ¬ (succ s)*(succ s) ≤ n*n + k,
@@ -88,8 +88,8 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
         have p : n*n ≤ n*n + k, from !le_add_right,
         have n : ¬ n*n ≤ n*n + k, by rewrite [-neqss at hg]; exact hg,
         absurd p n)
-     (λ sgen : s ≥ n,
+     (λ sgen : succ s > n,
-        by rewrite [sqrt_aux_succ_of_neg hg]; exact (sqrt_aux_offset_eq sgen)))
+        by rewrite [sqrt_aux_succ_of_neg hg]; exact (sqrt_aux_offset_eq (le_of_lt_succ sgen))))
 
 theorem sqrt_offset_eq {n k : nat} : k ≤ n + n → sqrt (n*n + k) = n :=
 assume h,

library/init/nat.lean

@@ -6,25 +6,43 @@ Authors: Floris van Doorn, Leonardo de Moura
 prelude
 import init.wf init.tactic init.num
 
-open eq.ops decidable
+open eq.ops decidable or
 
 namespace nat
   notation `ℕ` := nat
 
-  inductive lt (a : nat) : nat → Prop :=
+  /- basic definitions on natural numbers -/
-  | base : lt a (succ a)
+  inductive le (a : ℕ) : ℕ → Prop :=
-  | step : Π {b}, lt a b → lt a (succ b)
+  | refl : le a a
+  | step : Π {b}, le a b → le a (succ b)
 
-  notation a < b := lt a b
+  infix `≤` := le
+  attribute le.refl [refl]
 
-  definition le [reducible] (a b : nat) : Prop := a < succ b
+  definition lt [reducible] (n m : ℕ) := succ n ≤ m
+  definition ge [reducible] (n m : ℕ) := m ≤ n
+  definition gt [reducible] (n m : ℕ) := succ m ≤ n
+  infix `<` := lt
+  infix `≥` := ge
+  infix `>` := gt
 
-  notation a <= b := le a b
+  definition pred [unfold-c 1] (a : nat) : nat :=
-  notation a ≤ b := le a b
-
-  definition pred (a : nat) : nat :=
   nat.cases_on a zero (λ a₁, a₁)
 
+  -- add is defined in init.num
+
+  definition sub (a b : nat) : nat :=
+  nat.rec_on b a (λ b₁ r, pred r)
+
+  definition mul (a b : nat) : nat :=
+  nat.rec_on b zero (λ b₁ r, r + a)
+
+  notation a - b := sub a b
+  notation a * b := mul a b
+
+
+  /- properties of ℕ -/
+
   protected definition is_inhabited [instance] : inhabited nat :=
   inhabited.mk zero
 
@@ -35,27 +53,242 @@ namespace nat
   | has_decidable_eq (succ x) (succ y) :=
       match has_decidable_eq x y with
       | inl xeqy := inl (by rewrite xeqy)
-      | inr xney := inr (by intro h; injection h; contradiction)
+      | inr xney := inr (λ h : succ x = succ y, by injection h with xeqy; exact absurd xeqy xney)
       end
 
+  /- properties of inequality -/
+
+  theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
+
+  theorem le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
+
+  theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
+
+  theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
+  by induction H2 with n H2 IH;exact H1;exact le.step IH
+
+  theorem le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
+
+  theorem le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
+
+  theorem le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
+
+  theorem succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
+  by induction H;reflexivity;exact le.step v_0
+
+  theorem pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
+  by induction H;reflexivity;cases b;exact v_0;exact le.step v_0
+
+  theorem le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
+  pred_le_pred H
+
+  theorem le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
+  by cases n;exact le.step H;exact succ_le_succ H
+
+  theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
+  by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
+
+  theorem zero_le (n : ℕ) : 0 ≤ n :=
+  by induction n with n IH;apply le.refl;exact le.step IH
+
+  theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
+  le.step H
+
+  theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
+  by induction n with n IH;apply le.refl;exact le.step IH
+
+  theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
+  le.trans (le.step H1) H2
+
+  theorem lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
+  le.trans (succ_le_succ H1) H2
+
+  theorem lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
+  le.trans H1 H2
+
+  theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
+  begin
+    cases H1 with m' H1',
+    { reflexivity},
+    { cases H2 with n' H2',
+      { reflexivity},
+      { exfalso, apply not_succ_le_self, exact lt.trans H1' H2'}},
+  end
+
+  theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
+  by intro H; cases H
+
+  theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
+
+  theorem self_lt_succ (n : ℕ) : n < succ n := !le.refl
+  theorem lt.base (n : ℕ) : n < succ n := !le.refl
+
+  theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : false :=
+  !lt.irrefl (lt_of_le_of_lt H1 H2)
+
+  theorem lt_le_antisymm {n m : ℕ} (H1 : n < m) (H2 : m ≤ n) : false :=
+  le_lt_antisymm H2 H1
+
+  theorem lt.asymm {n m : ℕ} (H1 : n < m) (H2 : m < n) : false :=
+  le_lt_antisymm (le_of_lt H1) H2
+
+  definition lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
+  begin
+    revert b H1 H2 H3, induction a with a IH,
+    { intros, cases b,
+        exact H2 rfl,
+        exact H1 !zero_lt_succ},
+    { intros, cases b with b,
+        exact H3 !zero_lt_succ,
+      { apply IH,
+          intro H, exact H1 (succ_le_succ H),
+          intro H, exact H2 (congr rfl H),
+          intro H, exact H3 (succ_le_succ H)}}
+  end
+
+  theorem lt.trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
+  lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
+
+  definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
+  lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
+
+  theorem lt_or_ge (a b : ℕ) : (a < b) ∨ (a ≥ b) :=
+  lt_ge_by_cases inl inr
+
+  definition not_lt_zero (a : ℕ) : ¬ a < zero :=
+  by intro H; cases H
+
   -- less-than is well-founded
-  theorem lt.wf [instance] : well_founded lt :=
+  definition lt.wf [instance] : well_founded lt :=
-  well_founded.intro (λn, nat.rec_on n
+  begin
-    (acc.intro zero (λ (y : nat) (hlt : y < zero),
+    constructor, intro n, induction n with n IH,
-      have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
+    { constructor, intros n H, exfalso, exact !not_lt_zero H},
-        λ n₁ hlt, nat.lt.cases_on hlt
+    { constructor, intros m H,
-          (by contradiction)
+      assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
-          (by contradiction),
+        { intros n₁ hlt, induction hlt,
-      aux hlt rfl))
+          { intro p, injection p with q, exact q ▸ IH},
-    (λ (n : nat) (ih : acc lt n),
+          { intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
-      acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
+      apply aux H rfl},
-        have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
+  end
-          λ n₁ hlt, nat.lt.cases_on hlt
+
-            (λ (sn_eq_sm : succ n = succ m),
+  definition measure {A : Type} (f : A → ℕ) : A → A → Prop :=
-               by injection sn_eq_sm with neqm; rewrite neqm at ih; exact ih)
+  inv_image lt f
-            (λ b (hlt : m < b) (sn_eq_sb : succ n = succ b),
+
-               by injection sn_eq_sb with neqb; rewrite neqb at ih; exact acc.inv ih hlt),
+  definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
-        aux hlt rfl)))
+  inv_image.wf f lt.wf
+
+  theorem succ_lt_succ {a b : ℕ} (H : a < b) : succ a < succ b :=
+  succ_le_succ H
+
+  theorem lt_of_succ_lt {a b : ℕ} (H : succ a < b) : a < b :=
+  le_of_succ_le H
+
+  theorem lt_of_succ_lt_succ {a b : ℕ} (H : succ a < succ b) : a < b :=
+  le_of_succ_le_succ H
+
+  definition decidable_le [instance] : decidable_rel le :=
+  begin
+    intros n, induction n with n IH,
+    { intro m, left, apply zero_le},
+    { intro m, cases m with m,
+      { right, apply not_succ_le_zero},
+      { let H := IH m, clear IH,
+        cases H with H H,
+          left, exact succ_le_succ H,
+          right, intro H2, exact H (le_of_succ_le_succ H2)}}
+  end
+
+  definition decidable_lt [instance] : decidable_rel lt := _
+  definition decidable_gt [instance] : decidable_rel gt := _
+  definition decidable_ge [instance] : decidable_rel ge := _
+
+  theorem eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ∨ a < b :=
+  by cases H with b' H; exact inl rfl; exact inr (succ_le_succ H)
+
+  theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ∨ a < b) : a ≤ b :=
+  by cases H with H H; exact le_of_eq H; exact le_of_lt H
+
+  theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
+  or.rec_on (lt.trichotomy a b)
+    (λ hlt, absurd hlt hnlt)
+    (λ h, h)
+
+  theorem lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
+  succ_le_succ h
+
+  theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
+
+  theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
+
+  definition max (a b : ℕ) : ℕ := if a < b then b else a
+  definition min (a b : ℕ) : ℕ := if a < b then a else b
+
+  theorem max_self (a : ℕ) : max a a = a :=
+  eq.rec_on !if_t_t rfl
+
+  theorem max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
+  if_pos H
+
+  theorem max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
+  if_neg H
+
+  theorem eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
+  eq.rec_on (max_eq_right H) rfl
+
+  theorem eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
+  eq.rec_on (max_eq_left H) rfl
+
+  theorem le_max_left (a b : ℕ) : a ≤ max a b :=
+  by_cases
+    (λ h : a < b,   le_of_lt (eq.rec_on (eq_max_right h) h))
+    (λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
+
+  theorem le_max_right (a b : ℕ) : b ≤ max a b :=
+  by_cases
+    (λ h : a < b,   eq.rec_on (eq_max_right h) !le.refl)
+    (λ h : ¬ a < b, or.rec_on (eq_or_lt_of_not_lt h)
+      (λ heq, eq.rec_on heq (eq.rec_on (eq.symm (max_self a)) !le.refl))
+      (λ h : b < a,
+        have aux : a = max a b, from eq_max_left (lt.asymm h),
+        eq.rec_on aux (le_of_lt h)))
+
+  theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
+  by induction b with b IH;reflexivity; apply congr (eq.refl pred) IH
+
+  theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
+  eq.rec_on (succ_sub_succ_eq_sub a b) rfl
+
+  theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
+  nat.rec_on a
+    rfl
+    (λ a₁ (ih : zero - a₁ = zero), congr (eq.refl pred) ih)
+
+  theorem zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
+  eq.rec_on (zero_sub_eq_zero a) rfl
+
+  theorem sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
+  have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
+    λa h₁, le.rec_on h₁
+      (λb h₂, le.cases_on h₂
+        (lt.base zero)
+        (λ b₁ bpos,
+          eq.rec_on (sub_eq_succ_sub_succ zero b₁)
+           (eq.rec_on (zero_eq_zero_sub b₁) (lt.base zero))))
+      (λa₁ apos ih b h₂, le.cases_on h₂
+        (lt.base a₁)
+        (λ b₁ bpos,
+          eq.rec_on (sub_eq_succ_sub_succ a₁ b₁)
+            (lt.trans (@ih b₁ bpos) (lt.base a₁)))),
+  λ h₁ h₂, aux h₁ h₂
+
+  theorem sub_le (a b : ℕ) : a - b ≤ a :=
+  nat.rec_on b
+    (le.refl a)
+    (λ b₁ ih, le.trans !pred_le ih)
+
+end nat
+ exit
+
 
   definition measure {A : Type} (f : A → nat) : A → A → Prop :=
   inv_image lt f

tests/lean/protected_test.lean

@@ -2,10 +2,10 @@ namespace nat
   check induction_on      -- ERROR
   check rec_on            -- ERROR
   check nat.induction_on
-  check lt.rec_on         -- OK
+  check le.rec_on         -- OK
-  check nat.lt.rec_on
+  check nat.le.rec_on
-  namespace lt
+  namespace le
     check rec_on          -- ERROR
-    check lt.rec_on
+    check le.rec_on
-  end lt
+  end le
 end nat

tests/lean/protected_test.lean.expected.out

@@ -1,7 +1,7 @@
 protected_test.lean:2:8: error: unknown identifier 'induction_on'
 protected_test.lean:3:8: error: unknown identifier 'rec_on'
 nat.induction_on : ∀ (n : ℕ), ?C 0 → (∀ (a : ℕ), ?C a → ?C (succ a)) → ?C n
-lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
+le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1
-lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
+le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1
 protected_test.lean:8:10: error: unknown identifier 'rec_on'
-lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
+le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1

tests/lean/run/541a.lean

@@ -1,8 +1,6 @@
 import data.list data.nat
 open nat list eq.ops
 
-theorem nat.le_of_eq {x y : ℕ} (H : x = y) : x ≤ y := H ▸ !le.refl
-
 section
 
 variable {Q : Type}

tests/lean/run/constr_tac4.lean

@@ -4,17 +4,17 @@ open nat
 
 definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
 have aux : a < b → a < c, from
-  lt.rec_on H₂
+  le.rec_on H₂
     (λ h₁, lt.step h₁)
     (λ b₁ bb₁ ih h₁, by constructor; exact ih h₁),
 aux H₁
 
 definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
-lt.rec_on H
+le.rec_on H
   (by constructor)
   (λ b hlt ih, lt.trans ih (by constructor))
 
 definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
-lt.rec_on H
+le.rec_on H
   (by constructor; constructor)
   (λ b h ih, by constructor; exact ih)