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refactor(library): change mul.left_id to mul_one, and similarly for mul.right_id, add.left_id, add.right_id

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Jeremy Avigad 2014-12-23 21:14:19 -05:00
parent 5bc6dd84cf
commit 25394dddb7
17 changed files with 207 additions and 215 deletions
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46
library/algebra/group.lean
View file

@ -119,11 +119,11 @@ theorem add.right_cancel [s : add_right_cancel_semigroup A] {a b c : A} :
/- monoid -/
structure monoid [class] (A : Type) extends semigroup A, has_one A :=
(mul_left_id : ∀a, mul one a = a) (mul_right_id : ∀a, mul a one = a)
(one_mul : ∀a, mul one a = a) (mul_one : ∀a, mul a one = a)
theorem mul.left_id [s : monoid A] (a : A) : 1 * a = a := !monoid.mul_left_id
theorem one_mul [s : monoid A] (a : A) : 1 * a = a := !monoid.one_mul
theorem mul.right_id [s : monoid A] (a : A) : a * 1 = a := !monoid.mul_right_id
theorem mul_one [s : monoid A] (a : A) : a * 1 = a := !monoid.mul_one
structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A
@ -131,11 +131,11 @@ structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A
/- additive monoid -/
structure add_monoid [class] (A : Type) extends add_semigroup A, has_zero A :=
(add_left_id : ∀a, add zero a = a) (add_right_id : ∀a, add a zero = a)
(zero_add : ∀a, add zero a = a) (add_zero : ∀a, add a zero = a)
theorem add.left_id [s : add_monoid A] (a : A) : 0 + a = a := !add_monoid.add_left_id
theorem zero_add [s : add_monoid A] (a : A) : 0 + a = a := !add_monoid.zero_add
theorem add.right_id [s : add_monoid A] (a : A) : a + 0 = a := !add_monoid.add_right_id
theorem add_zero [s : add_monoid A] (a : A) : a + 0 = a := !add_monoid.add_zero
structure add_comm_monoid [class] (A : Type) extends add_monoid A, add_comm_semigroup A
@ -145,7 +145,7 @@ structure add_comm_monoid [class] (A : Type) extends add_monoid A, add_comm_semi
structure group [class] (A : Type) extends monoid A, has_inv A :=
(mul_left_inv : ∀a, mul (inv a) a = one)
-- Note: with more work, we could derive the axiom mul_left_id
-- Note: with more work, we could derive the axiom one_mul
section group
@ -158,21 +158,21 @@ section group
calc
a⁻¹ * (a * b) = a⁻¹ * a * b : !mul.assoc⁻¹
... = 1 * b : mul.left_inv
... = b : mul.left_id
... = b : one_mul
theorem inv_mul_cancel_right (a b : A) : a * b⁻¹ * b = a :=
calc
a * b⁻¹ * b = a * (b⁻¹ * b) : mul.assoc
... = a * 1 : mul.left_inv
... = a : mul.right_id
... = a : mul_one
theorem inv_eq_of_mul_eq_one {a b : A} (H : a * b = 1) : a⁻¹ = b :=
calc
a⁻¹ = a⁻¹ * 1 : !mul.right_id⁻¹
a⁻¹ = a⁻¹ * 1 : !mul_one⁻¹
... = a⁻¹ * (a * b) : H
... = b : inv_mul_cancel_left
theorem inv_one : 1⁻¹ = 1 := inv_eq_of_mul_eq_one (mul.left_id 1)
theorem inv_one : 1⁻¹ = 1 := inv_eq_of_mul_eq_one (one_mul 1)
theorem inv_inv (a : A) : (a⁻¹)⁻¹ = a := inv_eq_of_mul_eq_one (mul.left_inv a)
@ -203,13 +203,13 @@ section group
calc
a * (a⁻¹ * b) = a * a⁻¹ * b : !mul.assoc⁻¹
... = 1 * b : mul.right_inv
... = b : mul.left_id
... = b : one_mul
theorem mul_inv_cancel_right (a b : A) : a * b * b⁻¹ = a :=
calc
a * b * b⁻¹ = a * (b * b⁻¹) : mul.assoc
... = a * 1 : mul.right_inv
... = a : mul.right_id
... = a : mul_one
theorem inv_mul_eq (a b : A) : (a * b)⁻¹ = b⁻¹ * a⁻¹ :=
inv_eq_of_mul_eq_one
@ -222,7 +222,7 @@ section group
calc
a = a * b⁻¹ * b : !inv_mul_cancel_right⁻¹
... = 1 * b : H
... = b : mul.left_id
... = b : one_mul
-- TODO: better names for the next eight theorems? (Also for additive ones.)
theorem eq_mul_inv_of_mul_eq {a b c : A} (H : a * b = c) : a = c * b⁻¹ :=
@ -294,21 +294,21 @@ section add_group
calc
-a + (a + b) = -a + a + b : add.assoc
... = 0 + b : add.left_inv
... = b : add.left_id
... = b : zero_add
theorem neg_add_cancel_right (a b : A) : a + -b + b = a :=
calc
a + -b + b = a + (-b + b) : add.assoc
... = a + 0 : add.left_inv
... = a : add.right_id
... = a : add_zero
theorem neg_eq_of_add_eq_zero {a b : A} (H : a + b = 0) : -a = b :=
calc
-a = -a + 0 : add.right_id
-a = -a + 0 : add_zero
... = -a + (a + b) : H
... = b : neg_add_cancel_left
theorem neg_zero : -0 = 0 := neg_eq_of_add_eq_zero (add.left_id 0)
theorem neg_zero : -0 = 0 := neg_eq_of_add_eq_zero (zero_add 0)
theorem neg_neg (a : A) : -(-a) = a := neg_eq_of_add_eq_zero (add.left_inv a)
@ -338,13 +338,13 @@ section add_group
calc
a + (-a + b) = a + -a + b : add.assoc
... = 0 + b : add.right_inv
... = b : add.left_id
... = b : zero_add
theorem add_neg_cancel_right (a b : A) : a + b + -b = a :=
calc
a + b + -b = a + (b + -b) : add.assoc
... = a + 0 : add.right_inv
... = a : add.right_id
... = a : add_zero
theorem neg_add_eq (a b : A) : -(a + b) = -b + -a :=
neg_eq_of_add_eq_zero
@ -420,14 +420,14 @@ section add_group
calc
a = (a - b) + b : !sub_add_cancel⁻¹
... = 0 + b : H
... = b : add.left_id
... = b : zero_add
theorem eq_iff_sub_eq_zero (a b : A) : a = b ↔ a - b = 0 :=
iff.intro (assume H, H ▸ !sub_self) (assume H, eq_of_sub_eq_zero H)
theorem zero_sub_eq (a : A) : 0 - a = -a := !add.left_id
theorem zero_sub_eq (a : A) : 0 - a = -a := !zero_add
theorem sub_zero_eq (a : A) : a - 0 = a := subst (eq.symm neg_zero) !add.right_id
theorem sub_zero_eq (a : A) : a - 0 = a := subst (eq.symm neg_zero) !add_zero
theorem sub_neg_eq_add (a b : A) : a - (-b) = a + b := !neg_neg⁻¹ ▸ rfl

62
library/algebra/ordered_group.lean
View file

@ -88,28 +88,28 @@ section
-- here we start using properties of zero.
theorem add_nonneg {a b : A} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a + b :=
!add.left_id ▸ (add_le_add Ha Hb)
!zero_add ▸ (add_le_add Ha Hb)
theorem add_pos_of_pos_of_nonneg {a b : A} (Ha : 0 < a) (Hb : 0 ≤ b) : 0 < a + b :=
!add.left_id ▸ (add_lt_add_of_lt_of_le Ha Hb)
!zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
theorem add_pos_of_nonneg_of_pos {a b : A} (Ha : 0 ≤ a) (Hb : 0 < b) : 0 < a + b :=
!add.left_id ▸ (add_lt_add_of_le_of_lt Ha Hb)
!zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
theorem add_pos_of_pos_of_pos {a b : A} (Ha : 0 < a) (Hb : 0 < b) : 0 < a + b :=
!add.left_id ▸ (add_lt_add_of_lt_of_lt Ha Hb)
!zero_add ▸ (add_lt_add_of_lt_of_lt Ha Hb)
theorem add_nonpos {a b : A} (Ha : a ≤ 0) (Hb : b ≤ 0) : a + b ≤ 0 :=
!add.left_id ▸ (add_le_add Ha Hb)
!zero_add ▸ (add_le_add Ha Hb)
theorem add_neg_of_neg_of_nonpos {a b : A} (Ha : a < 0) (Hb : b ≤ 0) : a + b < 0 :=
!add.left_id ▸ (add_lt_add_of_lt_of_le Ha Hb)
!zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
theorem add_neg_of_nonpos_of_neg {a b : A} (Ha : a ≤ 0) (Hb : b < 0) : a + b < 0 :=
!add.left_id ▸ (add_lt_add_of_le_of_lt Ha Hb)
!zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
theorem add_neg_of_neg_of_neg {a b : A} (Ha : a < 0) (Hb : b < 0) : a + b < 0 :=
!add.left_id ▸ (add_lt_add_of_lt_of_lt Ha Hb)
!zero_add ▸ (add_lt_add_of_lt_of_lt Ha Hb)
-- TODO: add nonpos version (will be easier with simplifier)
theorem add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_noneng {a b : A}
@ -118,67 +118,67 @@ section
(assume Hab : a + b = 0,
have Ha' : a ≤ 0, from
calc
a = a + 0 : add.right_id
a = a + 0 : add_zero
... ≤ a + b : add_le_add_left Hb
... = 0 : Hab,
have Haz : a = 0, from le.antisym Ha' Ha,
have Hb' : b ≤ 0, from
calc
b = 0 + b : add.left_id
b = 0 + b : zero_add
... ≤ a + b : add_le_add_right Ha
... = 0 : Hab,
have Hbz : b = 0, from le.antisym Hb' Hb,
and.intro Haz Hbz)
(assume Hab : a = 0 ∧ b = 0,
(and.elim_left Hab)⁻¹ ▸ (and.elim_right Hab)⁻¹ ▸ (add.right_id 0))
(and.elim_left Hab)⁻¹ ▸ (and.elim_right Hab)⁻¹ ▸ (add_zero 0))
theorem le_add_of_nonneg_of_le (Ha : 0 ≤ a) (Hbc : b ≤ c) : b ≤ a + c :=
!add.left_id ▸ add_le_add Ha Hbc
!zero_add ▸ add_le_add Ha Hbc
theorem le_add_of_le_of_nonneg (Hbc : b ≤ c) (Ha : 0 ≤ a) : b ≤ c + a :=
!add.right_id ▸ add_le_add Hbc Ha
!add_zero ▸ add_le_add Hbc Ha
theorem lt_add_of_pos_of_le (Ha : 0 < a) (Hbc : b ≤ c) : b < a + c :=
!add.left_id ▸ add_lt_add_of_lt_of_le Ha Hbc
!zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
theorem lt_add_of_le_of_pos (Hbc : b ≤ c) (Ha : 0 < a) : b < c + a :=
!add.right_id ▸ add_lt_add_of_le_of_lt Hbc Ha
!add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
theorem add_le_of_nonpos_of_le (Ha : a ≤ 0) (Hbc : b ≤ c) : a + b ≤ c :=
!add.left_id ▸ add_le_add Ha Hbc
!zero_add ▸ add_le_add Ha Hbc
theorem add_le_of_le_of_nonpos (Hbc : b ≤ c) (Ha : a ≤ 0) : b + a ≤ c :=
!add.right_id ▸ add_le_add Hbc Ha
!add_zero ▸ add_le_add Hbc Ha
theorem add_lt_of_neg_of_le (Ha : a < 0) (Hbc : b ≤ c) : a + b < c :=
!add.left_id ▸ add_lt_add_of_lt_of_le Ha Hbc
!zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
theorem add_lt_of_le_of_neg (Hbc : b ≤ c) (Ha : a < 0) : b + a < c :=
!add.right_id ▸ add_lt_add_of_le_of_lt Hbc Ha
!add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
theorem lt_add_of_nonneg_of_lt (Ha : 0 ≤ a) (Hbc : b < c) : b < a + c :=
!add.left_id ▸ add_lt_add_of_le_of_lt Ha Hbc
!zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
theorem lt_add_of_lt_of_nonneg (Hbc : b < c) (Ha : 0 ≤ a) : b < c + a :=
!add.right_id ▸ add_lt_add_of_lt_of_le Hbc Ha
!add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
theorem lt_add_of_pos_of_lt (Ha : 0 < a) (Hbc : b < c) : b < a + c :=
!add.left_id ▸ add_lt_add_of_lt_of_lt Ha Hbc
!zero_add ▸ add_lt_add_of_lt_of_lt Ha Hbc
theorem lt_add_of_lt_of_pos (Hbc : b < c) (Ha : 0 < a) : b < c + a :=
!add.right_id ▸ add_lt_add_of_lt_of_lt Hbc Ha
!add_zero ▸ add_lt_add_of_lt_of_lt Hbc Ha
theorem add_lt_of_nonpos_of_lt (Ha : a ≤ 0) (Hbc : b < c) : a + b < c :=
!add.left_id ▸ add_lt_add_of_le_of_lt Ha Hbc
!zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
theorem add_lt_of_lt_of_nonpos (Hbc : b < c) (Ha : a ≤ 0) : b + a < c :=
!add.right_id ▸ add_lt_add_of_lt_of_le Hbc Ha
!add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
theorem add_lt_of_neg_of_lt (Ha : a < 0) (Hbc : b < c) : a + b < c :=
!add.left_id ▸ add_lt_add_of_lt_of_lt Ha Hbc
!zero_add ▸ add_lt_add_of_lt_of_lt Ha Hbc
theorem add_lt_of_lt_of_neg (Hbc : b < c) (Ha : a < 0) : b + a < c :=
!add.right_id ▸ add_lt_add_of_lt_of_lt Hbc Ha
!add_zero ▸ add_lt_add_of_lt_of_lt Hbc Ha
end
@ -194,7 +194,7 @@ structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_
definition ordered_comm_group.to_ordered_cancel_comm_monoid [instance] (A : Type)
[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
ordered_cancel_comm_monoid.mk ordered_comm_group.add ordered_comm_group.add_assoc
(@ordered_comm_group.zero A s) add.left_id add.right_id ordered_comm_group.add_comm
(@ordered_comm_group.zero A s) zero_add add_zero ordered_comm_group.add_comm
(@add.left_cancel _ _) (@add.right_cancel _ _)
has_le.le le.refl (@le.trans _ _) (@le.antisym _ _)
has_lt.lt (@lt_iff_le_and_ne _ _) ordered_comm_group.add_le_add_left
@ -212,8 +212,8 @@ section
theorem neg_le_neg_of_le {a b : A} (H : a ≤ b) : -b ≤ -a :=
have H1 : 0 ≤ -a + b, from !add.left_inv ▸ !(add_le_add_left H),
!add_neg_cancel_right ▸ !add.left_id ▸ add_le_add_right H1 (-b)
-- !add.left_id ▸ !add_neg_cancel_right ▸ add_le_add_right H1 (-b) -- doesn't work?
!add_neg_cancel_right ▸ !zero_add ▸ add_le_add_right H1 (-b)
-- !zero_add ▸ !add_neg_cancel_right ▸ add_le_add_right H1 (-b) -- doesn't work?
theorem neg_le_neg_iff_le : -a ≤ -b ↔ b ≤ a :=
iff.intro (take H, neg_neg a ▸ neg_neg b ▸ neg_le_neg_of_le H) neg_le_neg_of_le
@ -226,7 +226,7 @@ section
theorem neg_lt_neg_of_lt {a b : A} (H : a < b) : -b < -a :=
have H1 : 0 < -a + b, from !add.left_inv ▸ !(add_lt_add_left H),
!add_neg_cancel_right ▸ !add.left_id ▸ add_lt_add_right H1 (-b)
!add_neg_cancel_right ▸ !zero_add ▸ add_lt_add_right H1 (-b)
theorem neg_lt_neg_iff_lt : -a < -b ↔ b < a :=
iff.intro (take H, neg_neg a ▸ neg_neg b ▸ neg_lt_neg_of_lt H) neg_lt_neg_of_lt

14
library/algebra/ordered_ring.lean
View file

@ -155,8 +155,8 @@ structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A :
definition ordered_ring.to_ordered_semiring [instance] [s : ordered_ring A] : ordered_semiring A :=
ordered_semiring.mk ordered_ring.add ordered_ring.add_assoc !ordered_ring.zero
ordered_ring.add_left_id ordered_ring.add_right_id ordered_ring.add_comm ordered_ring.mul
ordered_ring.mul_assoc !ordered_ring.one ordered_ring.mul_left_id ordered_ring.mul_right_id
ordered_ring.zero_add ordered_ring.add_zero ordered_ring.add_comm ordered_ring.mul
ordered_ring.mul_assoc !ordered_ring.one ordered_ring.one_mul ordered_ring.mul_one
ordered_ring.left_distrib ordered_ring.right_distrib
zero_mul mul_zero !ordered_ring.zero_ne_one
(@add.left_cancel A _) (@add.right_cancel A _)
@ -263,15 +263,15 @@ section
(assume H : a ≥ 0, mul_nonneg_of_nonneg_of_nonneg H H)
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
theorem zero_le_one : 0 ≤ 1 := mul.left_id 1 ▸ mul_self_nonneg 1
theorem zero_le_one : 0 ≤ 1 := one_mul 1 ▸ mul_self_nonneg 1
theorem zero_lt_one : 0 < 1 := lt_of_le_of_ne zero_le_one zero_ne_one
-- TODO: move these to ordered_group.lean
theorem lt_add_of_pos_right {a b : A} (H : b > 0) : a < a + b := !add.right_id ▸ add_lt_add_left H a
theorem lt_add_of_pos_left {a b : A} (H : b > 0) : a < b + a := !add.left_id ▸ add_lt_add_right H a
theorem le_add_of_nonneg_right {a b : A} (H : b ≥ 0) : a ≤ a + b := !add.right_id ▸ add_le_add_left H a
theorem le_add_of_nonneg_left {a b : A} (H : b ≥ 0) : a ≤ b + a := !add.left_id ▸ add_le_add_right H a
theorem lt_add_of_pos_right {a b : A} (H : b > 0) : a < a + b := !add_zero ▸ add_lt_add_left H a
theorem lt_add_of_pos_left {a b : A} (H : b > 0) : a < b + a := !zero_add ▸ add_lt_add_right H a
theorem le_add_of_nonneg_right {a b : A} (H : b ≥ 0) : a ≤ a + b := !add_zero ▸ add_le_add_left H a
theorem le_add_of_nonneg_left {a b : A} (H : b ≥ 0) : a ≤ b + a := !zero_add ▸ add_le_add_right H a
-- TODO: remove after we short-circuit class-graph
definition linear_ordered_ring.to_mul [instance] [priority 100000] : has_mul A :=

24
library/algebra/ring.lean
View file

@ -86,7 +86,7 @@ section comm_semiring
theorem dvd.elim {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
exists.elim H₁ H₂
theorem dvd.refl : a | a := dvd.intro (!mul.right_id)
theorem dvd.refl : a | a := dvd.intro !mul_one
theorem dvd.trans {a b c : A} (H₁ : a | b) (H₂ : b | c) : a | c :=
dvd.elim H₁
@ -104,7 +104,7 @@ section comm_semiring
theorem dvd_zero : a | 0 := dvd.intro !mul_zero
theorem one_dvd : 1 | a := dvd.intro !mul.left_id
theorem one_dvd : 1 | a := dvd.intro !one_mul
theorem dvd_mul_right : a | a * b := dvd.intro rfl
@ -156,22 +156,22 @@ end comm_semiring
structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
definition ring.to_semiring [instance] [s : ring A] : semiring A :=
semiring.mk ring.add ring.add_assoc !ring.zero ring.add_left_id
add.right_id -- note: we've shown that add_right_id follows from add_left_id in add_comm_group
ring.add_comm ring.mul ring.mul_assoc !ring.one ring.mul_left_id ring.mul_right_id
semiring.mk ring.add ring.add_assoc !ring.zero ring.zero_add
add_zero -- note: we've shown that add_zero follows from zero_add in add_comm_group
ring.add_comm ring.mul ring.mul_assoc !ring.one ring.one_mul ring.mul_one
ring.left_distrib ring.right_distrib
(take a,
have H : 0 * a + 0 = 0 * a + 0 * a, from
calc
0 * a + 0 = 0 * a : add.right_id
... = (0 + 0) * a : {(add.right_id 0)⁻¹}
0 * a + 0 = 0 * a : add_zero
... = (0 + 0) * a : {(add_zero 0)⁻¹}
... = 0 * a + 0 * a : ring.right_distrib,
show 0 * a = 0, from (add.left_cancel H)⁻¹)
(take a,
have H : a * 0 + 0 = a * 0 + a * 0, from
calc
a * 0 + 0 = a * 0 : add.right_id
... = a * (0 + 0) : {(add.right_id 0)⁻¹}
a * 0 + 0 = a * 0 : add_zero
... = a * (0 + 0) : {(add_zero 0)⁻¹}
... = a * 0 + a * 0 : ring.left_distrib,
show a * 0 = 0, from (add.left_cancel H)⁻¹)
!ring.zero_ne_one
@ -232,8 +232,8 @@ structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
definition comm_ring.to_comm_semiring [instance] [s : comm_ring A] : comm_semiring A :=
comm_semiring.mk comm_ring.add comm_ring.add_assoc (@comm_ring.zero A s)
comm_ring.add_left_id comm_ring.add_right_id comm_ring.add_comm comm_ring.mul comm_ring.mul_assoc
(@comm_ring.one A s) comm_ring.mul_left_id comm_ring.mul_right_id comm_ring.left_distrib
comm_ring.zero_add comm_ring.add_zero comm_ring.add_comm comm_ring.mul comm_ring.mul_assoc
(@comm_ring.one A s) comm_ring.one_mul comm_ring.mul_one comm_ring.left_distrib
comm_ring.right_distrib zero_mul mul_zero (@comm_ring.zero_ne_one A s)
comm_ring.mul_comm
@ -246,7 +246,7 @@ section
theorem mul_self_sub_mul_self_eq : a * a - b * b = (a + b) * (a - b) := sorry
theorem mul_self_sub_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
mul.right_id 1 ▸ mul_self_sub_mul_self_eq a 1
mul_one 1 ▸ mul_self_sub_mul_self_eq a 1
end

28
library/data/int/basic.lean
View file

@ -220,12 +220,12 @@ or.elim (@le_or_gt n m)
H1⁻¹ ▸
(calc
m - n + n = m : add_sub_ge_left H
... = 0 + m : add.left_id))
... = 0 + m : zero_add))
(take H : m < n,
have H1 : repr (sub_nat_nat m n) = (0, succ (pred (n - m))), from sub_nat_nat_of_lt H ▸ rfl,
H1⁻¹ ▸
(calc
0 + n = n : add.left_id
0 + n = n : zero_add
... = n - m + m : add_sub_ge_left (lt_imp_le H)
... = succ (pred (n - m)) + m : (succ_pred_of_pos (sub_pos_of_gt H))⁻¹))
@ -336,7 +336,7 @@ cases_on a
from !repr_sub_nat_nat,
have H2 : padd (repr (of_nat m)) (repr (neg_succ_of_nat n')) = (m, 0 + succ n'),
from rfl,
(!add.left_id ▸ H2)⁻¹ ▸ H1))
(!zero_add ▸ H2)⁻¹ ▸ H1))
(take m',
cases_on b
(take n,
@ -344,7 +344,7 @@ cases_on a
from !repr_sub_nat_nat,
have H2 : padd (repr (neg_succ_of_nat m')) (repr (of_nat n)) = (0 + n, succ m'),
from rfl,
(!add.left_id ▸ H2)⁻¹ ▸ H1)
(!zero_add ▸ H2)⁻¹ ▸ H1)
(take n',!repr_sub_nat_nat))
theorem padd_congr {p p' q q' : ℕ × ℕ} (Ha : p ≡ p') (Hb : q ≡ q') : padd p q ≡ padd p' q' :=
@ -392,9 +392,9 @@ begin
apply H2
end
theorem add.right_id (a : ℤ) : a + 0 = a := cases_on a (take m, rfl) (take m', rfl)
theorem add_zero (a : ℤ) : a + 0 = a := cases_on a (take m, rfl) (take m', rfl)
theorem add.left_id (a : ℤ) : 0 + a = a := add.comm a 0 ▸ add.right_id a
theorem zero_add (a : ℤ) : 0 + a = a := add.comm a 0 ▸ add_zero a
/- negation -/
@ -500,13 +500,13 @@ cases_on a
pmul (repr (neg_succ_of_nat m')) (repr n) =
(0 * n + succ m' * 0, 0 * 0 + succ m' * n) : rfl
... = (0 + succ m' * 0, 0 * 0 + succ m' * n) : zero_mul
... = (0 + succ m' * 0, succ m' * n) : nat.add.left_id
... = (0 + succ m' * 0, succ m' * n) : nat.zero_add
... = repr (mul (neg_succ_of_nat m') n) : repr_neg_of_nat)⁻¹)
(take n',
(calc
pmul (repr (neg_succ_of_nat m')) (repr (neg_succ_of_nat n')) =
(0 + succ m' * succ n', 0 * succ n') : rfl
... = (succ m' * succ n', 0 * succ n') : nat.add.left_id
... = (succ m' * succ n', 0 * succ n') : nat.zero_add
... = (succ m' * succ n', 0) : zero_mul
... = repr (mul (neg_succ_of_nat m') (neg_succ_of_nat n')) : rfl)⁻¹))
@ -558,15 +558,15 @@ eq_of_repr_equiv_repr
... = pmul (repr a) (repr (b * c)) : repr_mul
... = repr (a * (b * c)) : repr_mul) ▸ !equiv.refl)
theorem mul.right_id (a : ℤ) : a * 1 = a :=
theorem mul_one (a : ℤ) : a * 1 = a :=
eq_of_repr_equiv_repr (equiv_of_eq
((calc
repr (a * 1) = pmul (repr a) (repr 1) : repr_mul
... = (pr1 (repr a), pr2 (repr a)) : by simp
... = repr a : prod.eta)))
theorem mul.left_id (a : ℤ) : 1 * a = a :=
mul.comm a 1 ▸ mul.right_id a
theorem one_mul (a : ℤ) : 1 * a = a :=
mul.comm a 1 ▸ mul_one a
theorem mul.right_distrib (a b c : ℤ) : (a + b) * c = a * c + b * c :=
eq_of_repr_equiv_repr
@ -601,8 +601,8 @@ or_of_or_of_imp_of_imp H3
(assume H : (nat_abs b) = nat.zero, nat_abs_eq_zero H)
definition integral_domain : algebra.integral_domain int :=
algebra.integral_domain.mk add add.assoc zero add.left_id add.right_id neg add.left_inv
add.comm mul mul.assoc (of_num 1) mul.left_id mul.right_id mul.left_distrib mul.right_distrib
algebra.integral_domain.mk add add.assoc zero zero_add add_zero neg add.left_inv
add.comm mul mul.assoc (of_num 1) one_mul mul_one mul.left_distrib mul.right_distrib
zero_ne_one mul.comm @eq_zero_or_eq_zero_of_mul_eq_zero
/-
@ -745,7 +745,7 @@ context port_algebra
end port_algebra
-- TODO: declare appropriate rewrite rules
-- add_rewrite add_left_id add_right_id
-- add_rewrite zero_add add_zero
-- add_rewrite add_comm add.assoc add_left_comm
-- add_rewrite sub_def add_inverse_right add_inverse_left
-- add_rewrite neg_add_distr

14
library/data/int/order.lean
View file

@ -45,7 +45,7 @@ exists.intro n (!add.comm ▸ iff.mp' !add_eq_iff_eq_add_neg (H1⁻¹))
-- ### partial order
theorem le_refl (a : ℤ) : a ≤ a :=
le_intro (add.right_id a)
le_intro (add_zero a)
theorem le_of_nat (n m : ℕ) : (of_nat n ≤ of_nat m) ↔ (n ≤ m) :=
iff.intro
@ -78,13 +78,13 @@ have H3 : a + of_nat (n + m) = a + 0, from
... = a + n + m : (add.assoc a n m)⁻¹
... = b + m : {Hn}
... = a : Hm
... = a + 0 : (add.right_id a)⁻¹,
... = a + 0 : (add_zero a)⁻¹,
have H4 : of_nat (n + m) = of_nat 0, from add.left_cancel H3,
have H5 : n + m = 0, from of_nat_inj H4,
have H6 : n = 0, from nat.eq_zero_of_add_eq_zero_right H5,
show a = b, from
calc
a = a + of_nat 0 : (add.right_id a)⁻¹
a = a + of_nat 0 : (add_zero a)⁻¹
... = a + n : {H6⁻¹}
... = b : Hn
@ -132,7 +132,7 @@ discriminate
(assume H2 : n = 0,
have H3 : a = b, from
calc
a = a + 0 : (add.right_id a)⁻¹
a = a + 0 : (add_zero a)⁻¹
... = a + n : {H2⁻¹}
... = b : Hn,
or.inr H3)
@ -188,7 +188,7 @@ theorem le_iff_sub_nonneg (a b : ℤ) : a ≤ b ↔ 0 ≤ b - a :=
iff.intro
(assume H, !sub_self ▸ sub_le_right H a)
(assume H,
have H1 : a ≤ b - a + a, from add.left_id a ▸ add_le_right H a,
have H1 : a ≤ b - a + a, from zero_add a ▸ add_le_right H a,
!neg_add_cancel_right ▸ H1)
@ -418,7 +418,7 @@ or_resolve_right (le_or_gt a b) H
theorem pos_imp_exists_nat {a : ℤ} (H : a ≥ 0) : ∃n : ℕ, a = n :=
obtain (n : ℕ) (Hn : of_nat 0 + n = a), from le_elim H,
exists.intro n (Hn⁻¹ ⬝ add.left_id n)
exists.intro n (Hn⁻¹ ⬝ zero_add n)
theorem neg_imp_exists_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -n :=
have H2 : -a ≥ 0, from zero_le_neg H,
@ -484,7 +484,7 @@ theorem mul_le_nonpos {a b c d : ℤ} (Ha : a ≤ 0) (Hb : b ≤ 0) (Hc : c ≤
le_trans (mul_le_right_nonpos Hb Hc) (mul_le_left_nonpos (le_trans Hc Ha) Hd)
theorem mul_lt_left_pos {a b c : ℤ} (Ha : a > 0) (H : b < c) : a * b < a * c :=
have H2 : a * b < a * b + a, from add.right_id (a * b) ▸ add_lt_left Ha (a * b),
have H2 : a * b < a * b + a, from add_zero (a * b) ▸ add_lt_left Ha (a * b),
have H3 : a * b + a ≤ a * c, from (by simp) ▸ mul_le_left_nonneg (lt_imp_le Ha) H,
lt_le_trans H2 H3

2
library/data/list/basic.lean
View file

@ -49,7 +49,7 @@ theorem length.nil : length (@nil T) = 0
theorem length.cons (x : T) (t : list T) : length (x::t) = succ (length t)
theorem length.append (s t : list T) : length (s ++ t) = length s + length t :=
induction_on s (!add.left_id⁻¹) (λx s H, !add.succ_left⁻¹ ▸ H ▸ rfl)
induction_on s (!zero_add⁻¹) (λx s H, !add.succ_left⁻¹ ▸ H ▸ rfl)
-- add_rewrite length_nil length_cons

46
library/data/nat/basic.lean
View file

@ -59,8 +59,6 @@ rfl
theorem pred.succ (n : ℕ) : pred (succ n) = n :=
rfl
irreducible pred
theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
induction_on n
(or.inl rfl)
@ -104,17 +102,15 @@ general m
/- addition -/
theorem add.right_id (n : ℕ) : n + 0 = n :=
theorem add_zero (n : ℕ) : n + 0 = n :=
rfl
theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
rfl
irreducible add
theorem add.left_id (n : ℕ) : 0 + n = n :=
theorem zero_add (n : ℕ) : 0 + n = n :=
induction_on n
!add.right_id
!add_zero
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ
@ -122,7 +118,7 @@ induction_on n
theorem add.succ_left (n m : ℕ) : (succ n) + m = succ (n + m) :=
induction_on m
(!add.right_id ▸ !add.right_id)
(!add_zero ▸ !add_zero)
(take k IH, calc
succ n + succ k = succ (succ n + k) : add_succ
... = succ (succ (n + k)) : IH
@ -130,7 +126,7 @@ induction_on m
theorem add.comm (n m : ℕ) : n + m = m + n :=
induction_on m
(!add.right_id ⬝ !add.left_id⁻¹)
(!add_zero ⬝ !zero_add⁻¹)
(take k IH, calc
n + succ k = succ (n+k) : add_succ
... = succ (k + n) : IH
@ -141,7 +137,7 @@ theorem succ_add_eq_add_succ (n m : ℕ) : succ n + m = n + succ m :=
theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
induction_on k
(!add.right_id ▸ !add.right_id)
(!add_zero ▸ !add_zero)
(take l IH,
calc
(n + m) + succ l = succ ((n + m) + l) : add_succ
@ -158,7 +154,7 @@ right_comm add.comm add.assoc n m k
theorem add.cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
induction_on n
(take H : 0 + m = 0 + k,
!add.left_id⁻¹ ⬝ H ⬝ !add.left_id)
!zero_add⁻¹ ⬝ H ⬝ !zero_add)
(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
have H2 : succ (n + m) = succ (n + k),
from calc
@ -190,10 +186,10 @@ theorem add.eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)
theorem add_one (n : ℕ) : n + 1 = succ n :=
!add.right_id ▸ !add_succ
!add_zero ▸ !add_succ
theorem one_add (n : ℕ) : 1 + n = succ n :=
!add.left_id ▸ !add.succ_left
!zero_add ▸ !add.succ_left
/- multiplication -/
@ -203,18 +199,16 @@ rfl
theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
rfl
irreducible mul
-- commutativity, distributivity, associativity, identity
theorem zero_mul (n : ℕ) : 0 * n = 0 :=
induction_on n
!mul_zero
(take m IH, !mul_succ ⬝ !add.right_id ⬝ IH)
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
induction_on m
(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add.right_id⁻¹)
(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
(take k IH, calc
succ n * succ k = succ n * k + succ n : mul_succ
... = n * k + k + succ n : IH
@ -236,7 +230,7 @@ theorem mul.right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
induction_on k
(calc
(n + m) * 0 = 0 : mul_zero
... = 0 + 0 : add.right_id
... = 0 + 0 : add_zero
... = n * 0 + 0 : mul_zero
... = n * 0 + m * 0 : mul_zero)
(take l IH, calc
@ -258,9 +252,7 @@ calc
theorem mul.assoc (n m k : ℕ) : (n * m) * k = n * (m * k) :=
induction_on k
(calc
(n * m) * 0 = 0 : mul_zero
... = n * 0 : mul_zero
... = n * (m * 0) : mul_zero)
(n * m) * 0 = n * (m * 0) : mul_zero)
(take l IH,
calc
(n * m) * succ l = (n * m) * l + n * m : mul_succ
@ -268,16 +260,16 @@ induction_on k
... = n * (m * l + m) : mul.left_distrib
... = n * (m * succ l) : mul_succ)
theorem mul.right_id (n : ℕ) : n * 1 = n :=
theorem mul_one (n : ℕ) : n * 1 = n :=
calc
n * 1 = n * 0 + n : mul_succ
... = 0 + n : mul_zero
... = n : add.left_id
... = n : zero_add
theorem mul.left_id (n : ℕ) : 1 * n = n :=
theorem one_mul (n : ℕ) : 1 * n = n :=
calc
1 * n = n * 1 : mul.comm
... = n : mul.right_id
... = n : mul_one
theorem eq_zero_or_eq_zero_of_mul_eq_zero {n m : ℕ} : n * m = 0 → n = 0 ∨ m = 0 :=
cases_on n
@ -299,8 +291,8 @@ section port_algebra
open algebra
protected definition comm_semiring [instance] : algebra.comm_semiring nat :=
algebra.comm_semiring.mk add add.assoc zero add.left_id add.right_id add.comm
mul mul.assoc (succ zero) mul.left_id mul.right_id mul.left_distrib mul.right_distrib
algebra.comm_semiring.mk add add.assoc zero zero_add add_zero add.comm
mul mul.assoc (succ zero) one_mul mul_one mul.left_distrib mul.right_distrib
zero_mul mul_zero (ne.symm (succ_ne_zero zero)) mul.comm
theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm

14
library/data/nat/div.lean
View file

@ -55,8 +55,8 @@ calc (x + z) div z
theorem div_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
induction_on y
(calc (x + zero * z) div z = (x + zero) div z : zero_mul
... = x div z : add.right_id
... = x div z + zero : add.right_id)
... = x div z : add_zero
... = x div z + zero : add_zero)
(take y,
assume IH : (x + y * z) div z = x div z + y, calc
(x + succ y * z) div z = (x + y * z + z) div z : by simp
@ -95,7 +95,7 @@ calc (x + z) mod z
theorem mod_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
induction_on y
(calc (x + zero * z) mod z = (x + zero) mod z : zero_mul
... = x mod z : add.right_id)
... = x mod z : add_zero)
(take y,
assume IH : (x + y * z) mod z = x mod z,
calc
@ -141,7 +141,7 @@ case_zero_pos y
(show x = x div 0 * 0 + x mod 0, from
(calc
x div 0 * 0 + x mod 0 = 0 + x mod 0 : mul_zero
... = x mod 0 : add.left_id
... = x mod 0 : zero_add
... = x : mod_zero)⁻¹)
(take y,
assume H : y > 0,
@ -264,7 +264,7 @@ theorem mod_eq_zero_imp_div_mul_eq {x y : ℕ} (H : x mod y = 0) : x div y * y =
(calc
x = x div y * y + x mod y : div_mod_eq
... = x div y * y + 0 : H
... = x div y * y : !add.right_id)⁻¹
... = x div y * y : !add_zero)⁻¹
-- add_rewrite dvd_imp_div_mul_eq
@ -291,7 +291,7 @@ show x mod y = 0, from
(assume ynz : y ≠ 0,
have ypos : y > 0, from ne_zero_imp_pos ynz,
have H3 : (z - x div y) * y < y, from H2⁻¹ ▸ mod_lt ypos,
have H4 : (z - x div y) * y < 1 * y, from !mul.left_id⁻¹ ▸ H3,
have H4 : (z - x div y) * y < 1 * y, from !one_mul⁻¹ ▸ H3,
have H5 : z - x div y < 1, from mul_lt_cancel_right H4,
have H6 : z - x div y = 0, from le_zero (lt_succ_imp_le H5),
calc
@ -323,7 +323,7 @@ case_zero_pos m
have H3 : n1 + n2 = 0, from eq_zero_of_zero_dvd H1,
have H4 : n1 = 0, from eq_zero_of_zero_dvd H2,
have H5 : n2 = 0, from calc
n2 = 0 + n2 : add.left_id
n2 = 0 + n2 : zero_add
... = n1 + n2 : H4
... = 0 : H3,
show 0 | n2, from H5 ▸ dvd.refl n2)

14
library/data/nat/order.lean
View file

@ -22,7 +22,7 @@ lt.step h
theorem le.add_right (n k : ℕ) : n ≤ n + k :=
induction_on k
(calc n ≤ n : le.refl n
... = n + zero : add.right_id)
... = n + zero : add_zero)
(λ k (ih : n ≤ n + k), calc
n ≤ succ (n + k) : le.succ_right ih
... = n + succ k : add_succ)
@ -47,7 +47,7 @@ theorem le_refl (n : ℕ) : n ≤ n :=
le.refl n
theorem zero_le (n : ℕ) : 0 ≤ n :=
le_intro !add.left_id
le_intro !zero_add
theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0 :=
obtain (k : ℕ) (Hk : n + k = 0), from le_elim H,
@ -70,10 +70,10 @@ have L1 : k + l = 0, from
n + (k + l) = n + k + l : !add.assoc⁻¹
... = m + l : {Hk}
... = n : Hl
... = n + 0 : !add.right_id⁻¹),
... = n + 0 : !add_zero⁻¹),
have L2 : k = 0, from eq_zero_of_add_eq_zero_right L1,
calc
n = n + 0 : !add.right_id⁻¹
n = n + 0 : !add_zero⁻¹
... = n + k : {L2⁻¹}
... = m : Hk
@ -137,7 +137,7 @@ discriminate
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : !add.right_id⁻¹
n = n + 0 : !add_zero⁻¹
... = n + k : {H3⁻¹}
... = m : Hk,
or.inr Heq)
@ -416,7 +416,7 @@ theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : exists l, n = succ l :=
lt_imp_eq_succ H
theorem add_pos_right {n k : ℕ} (H : k > 0) : n + k > n :=
!add.right_id ▸ add_lt_left H n
!add_zero ▸ add_lt_left H n
theorem add_pos_left {n : ℕ} {k : ℕ} (H : k > 0) : k + n > n :=
!add.comm ▸ add_pos_right H
@ -514,7 +514,7 @@ or.elim le_or_gt
show n = 1, from le_antisym H5 (succ_le_of_lt H3))
(assume H5 : n > 1,
have H6 : n * m ≥ 2 * 1, from mul_le (succ_le_of_lt H5) (succ_le_of_lt H4),
have H7 : 1 ≥ 2, from !mul.right_id ▸ H ▸ H6,
have H7 : 1 ≥ 2, from !mul_one ▸ H ▸ H6,
absurd !self_lt_succ (le_imp_not_gt H7))
theorem mul_eq_one_right {n m : ℕ} (H : n * m = 1) : m = 1 :=

22
library/data/nat/sub.lean
View file

@ -44,8 +44,8 @@ induction_on n !sub_zero_right (take k IH, !sub_succ_succ ⬝ IH)
theorem sub_add_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
induction_on k
(calc
(n + 0) - (m + 0) = n - (m + 0) : {!add.right_id}
... = n - m : {!add.right_id})
(n + 0) - (m + 0) = n - (m + 0) : {!add_zero}
... = n - m : {!add_zero})
(take l : nat,
assume IH : (n + l) - (m + l) = n - m,
calc
@ -59,7 +59,7 @@ theorem sub_add_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
theorem sub_add_left (n m : ℕ) : n + m - m = n :=
induction_on m
(!add.right_id⁻¹ ▸ !sub_zero_right)
(!add_zero⁻¹ ▸ !sub_zero_right)
(take k : ℕ,
assume IH : n + k - k = n,
calc
@ -75,7 +75,7 @@ theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) :=
induction_on k
(calc
n - m - 0 = n - m : !sub_zero_right
... = n - (m + 0) : {!add.right_id⁻¹})
... = n - (m + 0) : {!add_zero⁻¹})
(take l : nat,
assume IH : n - m - l = n - (m + l),
calc
@ -184,7 +184,7 @@ sub_induction n m
(take k,
assume H : 0 ≤ k,
calc
0 + (k - 0) = k - 0 : !add.left_id
0 + (k - 0) = k - 0 : !zero_add
... = k : !sub_zero_right)
(take k, assume H : succ k ≤ 0, absurd H !not_succ_zero_le)
(take k l,
@ -201,7 +201,7 @@ theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n :=
theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n :=
calc
n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
... = n : !add.right_id
... = n : !add_zero
theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m :=
!add.comm ▸ add_sub_ge
@ -254,7 +254,7 @@ sub_split
(take k : ℕ,
assume H1 : m + k = n,
assume H2 : k = 0,
have H3 : n = m, from !add.right_id ▸ H2 ▸ H1⁻¹,
have H3 : n = m, from !add_zero ▸ H2 ▸ H1⁻¹,
H3 ▸ !le_refl)
theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
@ -308,7 +308,7 @@ sub_split
theorem sub_pos_of_gt {m n : ℕ} (H : n > m) : n - m > 0 :=
have H1 : n = n - m + m, from (add_sub_ge_left (lt_imp_le H))⁻¹,
have H2 : 0 + m < n - m + m, from (add.left_id m)⁻¹ ▸ H1 ▸ H,
have H2 : 0 + m < n - m + m, from (zero_add m)⁻¹ ▸ H1 ▸ H,
!add_lt_cancel_right H2
-- theorem sub_lt_cancel_right {n m k : ℕ) (H : n - k < m - k) : n < m
@ -321,14 +321,14 @@ have H2 : 0 + m < n - m + m, from (add.left_id m)⁻¹ ▸ H1 ▸ H,
theorem sub_triangle_inequality (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
sub_split
(assume H : n ≤ m, !add.left_id⁻¹ ▸ sub_le_right H k)
(assume H : n ≤ m, !zero_add⁻¹ ▸ sub_le_right H k)
(take mn : ℕ,
assume Hmn : m + mn = n,
sub_split
(assume H : m ≤ k,
have H2 : n - k ≤ n - m, from sub_le_left H n,
have H3 : n - k ≤ mn, from sub_intro Hmn ▸ H2,
show n - k ≤ mn + 0, from !add.right_id⁻¹ ▸ H3)
show n - k ≤ mn + 0, from !add_zero⁻¹ ▸ H3)
(take km : ℕ,
assume Hkm : k + km = m,
have H : k + (mn + km) = n, from
@ -372,7 +372,7 @@ le_antisym H3 H5
theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n :=
calc
dist n m = 0 + (m - n) : {le_imp_sub_eq_zero H}
... = m - n : !add.left_id
... = m - n : !zero_add
theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
!dist_comm ▸ dist_le H

2
tests/lean/run/algebra3.lean
View file

@ -39,7 +39,7 @@ structure ring [class] (A : Type)
assoc→add.assoc
comm→add.comm
one→zero
right_id→add.right_id
right_id→add_zero
left_id→add.left_id
inv→uminus
is_inv→uminus_is_inv,

2
tests/lean/run/imp_curly.lean
View file

@ -3,7 +3,7 @@ open nat
theorem zero_left (n : ℕ) : 0 + n = n :=
nat.induction_on n
!add.right_id
!add_zero
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ

2
tests/lean/run/nat_bug5.lean
View file

@ -13,7 +13,7 @@ infixl `*` := mul
axiom add_one (n:nat) : n + (succ zero) = succ n
axiom mul_zero_right (n : nat) : n * zero = zero
axiom add.right_id (n : nat) : n + zero = n
axiom add_zero (n : nat) : n + zero = n
axiom mul_succ_right (n m : nat) : n * succ m = n * m + n
axiom add_assoc (n m k : nat) : (n + m) + k = n + (m + k)
axiom add_right_comm (n m k : nat) : n + m + k = n + k + m

2
tests/lean/slow/list_elab2.lean
View file

@ -110,7 +110,7 @@ theorem length_concat (s t : list T) : length (s ++ t) = length s + length t :=
list_induction_on s
(calc
length (concat nil t) = length t : refl _
... = 0 + length t : {symm !add.left_id}
... = 0 + length t : {symm !zero_add}
... = length (@nil T) + length t : refl _)
(take x s,
assume H : length (concat s t) = length s + length t,

64
tests/lean/slow/nat_bug2.lean
View file

@ -132,13 +132,13 @@ theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
-------------------------------------------------- add
definition add (x y : ℕ) : ℕ := plus x y
infixl `+` := add
theorem add.right_id (n : ℕ) : n + 0 = n
theorem add_zero (n : ℕ) : n + 0 = n
theorem add_succ (n m : ℕ) : n + succ m = succ (n + m)
---------- comm, assoc
theorem add.left_id (n : ℕ) : 0 + n = n
theorem zero_add (n : ℕ) : 0 + n = n
:= induction_on n
(add.right_id 0)
(add_zero 0)
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ _ _
@ -147,8 +147,8 @@ theorem add.left_id (n : ℕ) : 0 + n = n
theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
:= induction_on m
(calc
succ n + 0 = succ n : add.right_id (succ n)
... = succ (n + 0) : {symm (add.right_id n)})
succ n + 0 = succ n : add_zero (succ n)
... = succ (n + 0) : {symm (add_zero n)})
(take k IH,
calc
succ n + succ k = succ (succ n + k) : add_succ _ _
@ -157,7 +157,7 @@ theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
theorem add_comm (n m : ℕ) : n + m = m + n
:= induction_on m
(trans (add.right_id _) (symm (add.left_id _)))
(trans (add_zero _) (symm (zero_add _)))
(take k IH,
calc
n + succ k = succ (n+k) : add_succ _ _
@ -177,8 +177,8 @@ theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n
theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k)
:= induction_on k
(calc
(n + m) + 0 = n + m : add.right_id _
... = n + (m + 0) : {symm (add.right_id m)})
(n + m) + 0 = n + m : add_zero _
... = n + (m + 0) : {symm (add_zero m)})
(take l IH,
calc
(n + m) + succ l = succ ((n + m) + l) : add_succ _ _
@ -200,9 +200,9 @@ theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k
induction_on n
(take H : 0 + m = 0 + k,
calc
m = 0 + m : symm (add.left_id m)
m = 0 + m : symm (zero_add m)
... = 0 + k : H
... = k : add.left_id k)
... = k : zero_add k)
(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
have H2 : succ (n + m) = succ (n + k),
from calc
@ -249,13 +249,13 @@ theorem add_one (n:ℕ) : n + 1 = succ n
:=
calc
n + 1 = succ (n + 0) : add_succ _ _
... = succ n : {add.right_id _}
... = succ n : {add_zero _}
theorem add_one_left (n:ℕ) : 1 + n = succ n
:=
calc
1 + n = succ (0 + n) : succ_add _ _
... = succ n : {add.left_id _}
... = succ n : {zero_add _}
--the following theorem has a terrible name, but since the name is not a substring or superstring of another name, it is at least easy to globally replace it
theorem induction_plus_one {P : ℕ → Prop} (a : ℕ) (H1 : P 0)
@ -278,7 +278,7 @@ theorem mul_zero_left (n:ℕ) : 0 * n = 0
(take m IH,
calc
0 * succ m = 0 * m + 0 : mul_succ_right _ _
... = 0 * m : add.right_id _
... = 0 * m : add_zero _
... = 0 : IH)
theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
@ -286,7 +286,7 @@ theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
(calc
succ n * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * 0 + 0 : symm (add.right_id _))
... = n * 0 + 0 : symm (add_zero _))
(take k IH,
calc
succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
@ -309,7 +309,7 @@ theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k
:= induction_on k
(calc
(n + m) * 0 = 0 : mul_zero_right _
... = 0 + 0 : symm (add.right_id _)
... = 0 + 0 : symm (add_zero _)
... = n * 0 + 0 : refl _
... = n * 0 + m * 0 : refl _)
(take l IH, calc
@ -351,7 +351,7 @@ theorem mul_one_right (n : ℕ) : n * 1 = n
:= calc
n * 1 = n * 0 + n : mul_succ_right n 0
... = 0 + n : {mul_zero_right n}
... = n : add.left_id n
... = n : zero_add n
theorem mul_one_left (n : ℕ) : 1 * n = n
:= calc
@ -397,10 +397,10 @@ theorem le_intro2 (n m : ℕ) : n ≤ n + m
:= le_intro (refl (n + m))
theorem le_refl (n : ℕ) : n ≤ n
:= le_intro (add.right_id n)
:= le_intro (add_zero n)
theorem zero_le (n : ℕ) : 0 ≤ n
:= le_intro (add.left_id n)
:= le_intro (zero_add n)
theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0
:=
@ -434,10 +434,10 @@ theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m
n + (k + l) = n + k + l : { symm (add_assoc n k l) }
... = m + l : { Hk }
... = n : Hl
... = n + 0 : symm (add.right_id n)),
... = n + 0 : symm (add_zero n)),
have L2 : k = 0, from eq_zero_of_add_eq_zero_right L1,
calc
n = n + 0 : symm (add.right_id n)
n = n + 0 : symm (add_zero n)
... = n + k : { symm L2 }
... = m : Hk
@ -487,7 +487,7 @@ theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : (add.right_id n)⁻¹
n = n + 0 : (add_zero n)⁻¹
... = n + k : {H3⁻¹}
... = m : Hk,
or_intro_right _ Heq)
@ -574,7 +574,7 @@ theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : symm (add.right_id n)
n = n + 0 : symm (add_zero n)
... = n + k : {symm H3}
... = m : Hk,
or_intro_right _ Heq)
@ -792,7 +792,7 @@ theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
from calc
m = k + l : symm Hl
... = k + 0 : {H2}
... = k : add.right_id k,
... = k : add_zero k,
have H4 : m < succ k, from subst H3 (lt_self_succ m),
or_intro_right _ H4)
(take l2 : ℕ,
@ -911,7 +911,7 @@ theorem succ_imp_pos {n m : ℕ} (H : n = succ m) : n > 0
:= subst (symm H) (succ_pos m)
theorem add_pos_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
:= subst (add.right_id n) (add_lt_left H n)
:= subst (add_zero n) (add_lt_left H n)
theorem add_pos_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n
:= subst (add_comm n k) (add_pos_right n H)
@ -1100,8 +1100,8 @@ theorem sub_self (n : ℕ) : n - n = 0
theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m
:= induction_on k
(calc
(n + 0) - (m + 0) = n - (m + 0) : {add.right_id _}
... = n - m : {add.right_id _})
(n + 0) - (m + 0) = n - (m + 0) : {add_zero _}
... = n - m : {add_zero _})
(take l : ℕ,
assume IH : (n + l) - (m + l) = n - m,
calc
@ -1115,7 +1115,7 @@ theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m
theorem sub_add_left (n m : ℕ) : n + m - m = n
:= induction_on m
(subst (symm (add.right_id n)) (sub_zero_right n))
(subst (symm (add_zero n)) (sub_zero_right n))
(take k : ℕ,
assume IH : n + k - k = n,
calc
@ -1127,7 +1127,7 @@ theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k)
:= induction_on k
(calc
n - m - 0 = n - m : sub_zero_right _
... = n - (m + 0) : {symm (add.right_id m)})
... = n - (m + 0) : {symm (add_zero m)})
(take l : ℕ,
assume IH : n - m - l = n - (m + l),
calc
@ -1229,7 +1229,7 @@ theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m
(take k,
assume H : 0 ≤ k,
calc
0 + (k - 0) = k - 0 : add.left_id (k - 0)
0 + (k - 0) = k - 0 : zero_add (k - 0)
... = k : sub_zero_right k)
(take k, assume H : succ k ≤ 0, absurd H (not_succ_zero_le k))
(take k l,
@ -1246,7 +1246,7 @@ theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n
theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n
:= calc
n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
... = n : add.right_id n
... = n : add_zero n
theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m
:= subst (add_comm m (n - m)) add_sub_ge
@ -1306,7 +1306,7 @@ theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m
(take k : ℕ,
assume H1 : m + k = n,
assume H2 : k = 0,
have H3 : n = m, from subst (add.right_id m) (subst H2 (symm H1)),
have H3 : n = m, from subst (add_zero m) (subst H2 (symm H1)),
subst H3 (le_refl n))
theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
@ -1372,7 +1372,7 @@ theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n
:= calc
dist n m = (n - m) + (m - n) : refl _
... = 0 + (m - n) : {le_imp_sub_eq_zero H}
... = m - n : add.left_id (m - n)
... = m - n : zero_add (m - n)
theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m
:= subst (dist_comm m n) (dist_le H)

64
tests/lean/slow/nat_wo_hints.lean
View file

@ -126,13 +126,13 @@ theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
-------------------------------------------------- add
definition add (x y : ℕ) : ℕ := plus x y
infixl `+` := add
theorem add.right_id (n : ℕ) : n + 0 = n
theorem add_zero (n : ℕ) : n + 0 = n
theorem add_succ (n m : ℕ) : n + succ m = succ (n + m)
---------- comm, assoc
theorem add.left_id (n : ℕ) : 0 + n = n
theorem zero_add (n : ℕ) : 0 + n = n
:= induction_on n
(add.right_id 0)
(add_zero 0)
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ _ _
@ -141,8 +141,8 @@ theorem add.left_id (n : ℕ) : 0 + n = n
theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
:= induction_on m
(calc
succ n + 0 = succ n : add.right_id (succ n)
... = succ (n + 0) : {symm (add.right_id n)})
succ n + 0 = succ n : add_zero (succ n)
... = succ (n + 0) : {symm (add_zero n)})
(take k IH,
calc
succ n + succ k = succ (succ n + k) : add_succ _ _
@ -151,7 +151,7 @@ theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
theorem add_comm (n m : ℕ) : n + m = m + n
:= induction_on m
(trans (add.right_id _) (symm (add.left_id _)))
(trans (add_zero _) (symm (zero_add _)))
(take k IH,
calc
n + succ k = succ (n+k) : add_succ _ _
@ -171,8 +171,8 @@ theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n
theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k)
:= induction_on k
(calc
(n + m) + 0 = n + m : add.right_id _
... = n + (m + 0) : {symm (add.right_id m)})
(n + m) + 0 = n + m : add_zero _
... = n + (m + 0) : {symm (add_zero m)})
(take l IH,
calc
(n + m) + succ l = succ ((n + m) + l) : add_succ _ _
@ -194,9 +194,9 @@ theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k
induction_on n
(take H : 0 + m = 0 + k,
calc
m = 0 + m : symm (add.left_id m)
m = 0 + m : symm (zero_add m)
... = 0 + k : H
... = k : add.left_id k)
... = k : zero_add k)
(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
have H2 : succ (n + m) = succ (n + k),
from calc
@ -243,13 +243,13 @@ theorem add_one (n:ℕ) : n + 1 = succ n
:=
calc
n + 1 = succ (n + 0) : add_succ _ _
... = succ n : {add.right_id _}
... = succ n : {add_zero _}
theorem add_one_left (n:ℕ) : 1 + n = succ n
:=
calc
1 + n = succ (0 + n) : succ_add _ _
... = succ n : {add.left_id _}
... = succ n : {zero_add _}
--the following theorem has a terrible name, but since the name is not a substring or superstring of another name, it is at least easy to globally replace it
theorem induction_plus_one {P : ℕ → Prop} (a : ℕ) (H1 : P 0)
@ -272,7 +272,7 @@ theorem mul_zero_left (n:ℕ) : 0 * n = 0
(take m IH,
calc
0 * succ m = 0 * m + 0 : mul_succ_right _ _
... = 0 * m : add.right_id _
... = 0 * m : add_zero _
... = 0 : IH)
theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
@ -280,7 +280,7 @@ theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
(calc
succ n * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * 0 + 0 : symm (add.right_id _))
... = n * 0 + 0 : symm (add_zero _))
(take k IH,
calc
succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
@ -303,7 +303,7 @@ theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k
:= induction_on k
(calc
(n + m) * 0 = 0 : mul_zero_right _
... = 0 + 0 : symm (add.right_id _)
... = 0 + 0 : symm (add_zero _)
... = n * 0 + 0 : eq.refl _
... = n * 0 + m * 0 : eq.refl _)
(take l IH, calc
@ -345,7 +345,7 @@ theorem mul_one_right (n : ℕ) : n * 1 = n
:= calc
n * 1 = n * 0 + n : mul_succ_right n 0
... = 0 + n : {mul_zero_right n}
... = n : add.left_id n
... = n : zero_add n
theorem mul_one_left (n : ℕ) : 1 * n = n
:= calc
@ -391,10 +391,10 @@ theorem le_intro2 (n m : ℕ) : n ≤ n + m
:= le_intro (eq.refl (n + m))
theorem le_refl (n : ℕ) : n ≤ n
:= le_intro (add.right_id n)
:= le_intro (add_zero n)
theorem zero_le (n : ℕ) : 0 ≤ n
:= le_intro (add.left_id n)
:= le_intro (zero_add n)
theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0
:=
@ -428,10 +428,10 @@ theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m
n + (k + l) = n + k + l : { symm (add_assoc n k l) }
... = m + l : { Hk }
... = n : Hl
... = n + 0 : symm (add.right_id n)),
... = n + 0 : symm (add_zero n)),
have L2 : k = 0, from eq_zero_of_add_eq_zero_right L1,
calc
n = n + 0 : symm (add.right_id n)
n = n + 0 : symm (add_zero n)
... = n + k : { symm L2 }
... = m : Hk
@ -481,7 +481,7 @@ theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : (add.right_id n)⁻¹
n = n + 0 : (add_zero n)⁻¹
... = n + k : {H3⁻¹}
... = m : Hk,
or.intro_right _ Heq)
@ -568,7 +568,7 @@ theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : symm (add.right_id n)
n = n + 0 : symm (add_zero n)
... = n + k : {symm H3}
... = m : Hk,
or.intro_right _ Heq)
@ -790,7 +790,7 @@ theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
from calc
m = k + l : symm Hl
... = k + 0 : {H2}
... = k : add.right_id k,
... = k : add_zero k,
have H4 : m < succ k, from subst H3 (lt_self_succ m),
or.intro_right _ H4)
(take l2 : ℕ,
@ -915,7 +915,7 @@ theorem ne_zero_pos {n : ℕ} (H : n ≠ 0) : n > 0
:= or.elim (zero_or_pos n) (take H2 : n = 0, absurd H2 H) (take H2 : n > 0, H2)
theorem add_pos_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
:= subst (add.right_id n) (add_lt_left H n)
:= subst (add_zero n) (add_lt_left H n)
theorem add_pos_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n
:= subst (add_comm n k) (add_pos_right n H)
@ -1104,8 +1104,8 @@ theorem sub_self (n : ℕ) : n - n = 0
theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m
:= induction_on k
(calc
(n + 0) - (m + 0) = n - (m + 0) : {add.right_id _}
... = n - m : {add.right_id _})
(n + 0) - (m + 0) = n - (m + 0) : {add_zero _}
... = n - m : {add_zero _})
(take l : ℕ,
assume IH : (n + l) - (m + l) = n - m,
calc
@ -1119,7 +1119,7 @@ theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m
theorem sub_add_left (n m : ℕ) : n + m - m = n
:= induction_on m
(subst (symm (add.right_id n)) (sub_zero_right n))
(subst (symm (add_zero n)) (sub_zero_right n))
(take k : ℕ,
assume IH : n + k - k = n,
calc
@ -1131,7 +1131,7 @@ theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k)
:= induction_on k
(calc
n - m - 0 = n - m : sub_zero_right _
... = n - (m + 0) : {symm (add.right_id m)})
... = n - (m + 0) : {symm (add_zero m)})
(take l : ℕ,
assume IH : n - m - l = n - (m + l),
calc
@ -1233,7 +1233,7 @@ theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m
(take k,
assume H : 0 ≤ k,
calc
0 + (k - 0) = k - 0 : add.left_id (k - 0)
0 + (k - 0) = k - 0 : zero_add (k - 0)
... = k : sub_zero_right k)
(take k, assume H : succ k ≤ 0, absurd H (not_succ_zero_le k))
(take k l,
@ -1250,7 +1250,7 @@ theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n
theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n
:= calc
n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
... = n : add.right_id n
... = n : add_zero n
theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m
:= subst (add_comm m (n - m)) add_sub_ge
@ -1310,7 +1310,7 @@ theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m
(take k : ℕ,
assume H1 : m + k = n,
assume H2 : k = 0,
have H3 : n = m, from subst (add.right_id m) (subst H2 (symm H1)),
have H3 : n = m, from subst (add_zero m) (subst H2 (symm H1)),
subst H3 (le_refl n))
theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
@ -1376,7 +1376,7 @@ theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n
:= calc
dist n m = (n - m) + (m - n) : eq.refl _
... = 0 + (m - n) : {le_imp_sub_eq_zero H}
... = m - n : add.left_id (m - n)
... = m - n : zero_add (m - n)
theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m
:= subst (dist_comm m n) (dist_le H)