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feat/refactor(library/data/int): revise and add theorems

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This commit is contained in:
Jeremy Avigad 2015-02-25 14:14:57 -05:00 committed by Leonardo de Moura
parent cfdaffb6f5
commit 7a57908d62
4 changed files with 86 additions and 25 deletions
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2
library/algebra/group.lean
View file

@ -373,6 +373,8 @@ section add_group
infix `-` := sub
theorem sub_eq_add_neg (a b : A) : a - b = a + -b := rfl
theorem sub_self (a : A) : a - a = 0 := !add.right_inv
theorem sub_add_cancel (a b : A) : a - b + b = a := !neg_add_cancel_right

12
library/algebra/ordered_group.lean
View file

@ -370,6 +370,18 @@ section
theorem sub_lt_sub_of_lt_of_le {a b c d : A} (Hab : a < b) (Hcd : c ≤ d) : a - d < b - c :=
add_lt_add_of_lt_of_le Hab (neg_le_neg Hcd)
theorem sub_le_self (a : A) {b : A} (H : b ≥ 0) : a - b ≤ a :=
calc
a - b = a + -b : rfl
... ≤ a + 0 : add_le_add_left (neg_nonpos_of_nonneg H)
... = a : add_zero
theorem sub_lt_self (a : A) {b : A} (H : b > 0) : a - b < a :=
calc
a - b = a + -b : rfl
... < a + 0 : add_lt_add_left (neg_neg_of_pos H)
... = a : add_zero
end
structure decidable_linear_ordered_comm_group [class] (A : Type)

24
library/data/int/basic.lean
View file

@ -93,32 +93,34 @@ infix * := int.mul
/- some basic functions and properties -/
theorem of_nat_inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
theorem of_nat.inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
int.no_confusion H (λe, e)
theorem neg_succ_of_nat_inj {m n : ℕ} (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
theorem neg_succ_of_nat.inj {m n : ℕ} (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
int.no_confusion H (λe, e)
theorem neg_succ_of_nat_eq (n : ℕ) : -[n +1] = -(n + 1) := rfl
definition has_decidable_eq [instance] : decidable_eq ℤ :=
take a b,
int.cases_on a
(take m,
int.cases_on b
(take n,
if H : m = n then inl (congr_arg of_nat H) else inr (take H1, H (of_nat_inj H1)))
if H : m = n then inl (congr_arg of_nat H) else inr (take H1, H (of_nat.inj H1)))
(take n', inr (assume H, int.no_confusion H)))
(take m',
int.cases_on b
(take n, inr (assume H, int.no_confusion H))
(take n',
(if H : m' = n' then inl (congr_arg neg_succ_of_nat H) else
inr (take H1, H (neg_succ_of_nat_inj H1)))))
inr (take H1, H (neg_succ_of_nat.inj H1)))))
theorem add_of_nat (n m : nat) : of_nat n + of_nat m = #nat n + m := rfl
theorem of_nat_add_of_nat (n m : nat) : of_nat n + of_nat m = #nat n + m := rfl
theorem of_nat_succ (n : ℕ) : of_nat (succ n) = of_nat n + 1 := rfl
theorem mul_of_nat (n m : ℕ) : of_nat n * of_nat m = n * m := rfl
theorem of_nat_mul_of_nat (n m : ℕ) : of_nat n * of_nat m = n * m := rfl
theorem sub_nat_nat_of_ge {m n : ℕ} (H : m ≥ n) : sub_nat_nat m n = of_nat (m - n) :=
have H1 : n - m = 0, from sub_eq_zero_of_le H,
@ -593,7 +595,7 @@ calc
theorem zero_ne_one : (typeof 0 : int) ≠ 1 :=
assume H : 0 = 1,
show false, from succ_ne_zero 0 ((of_nat_inj H)⁻¹)
show false, from succ_ne_zero 0 ((of_nat.inj H)⁻¹)
theorem eq_zero_or_eq_zero_of_mul_eq_zero {a b : ℤ} (H : a * b = 0) : a = 0 ∨ b = 0 :=
have H2 : (nat_abs a) * (nat_abs b) = nat.zero, from
@ -678,6 +680,7 @@ section port_algebra
@algebra.add_eq_iff_eq_add_neg _ _
definition sub (a b : ℤ) : ℤ := algebra.sub a b
infix - := int.sub
theorem sub_eq_add_neg : ∀a b : ℤ, a - b = a + -b := algebra.sub_eq_add_neg
theorem sub_self : ∀a : ℤ, a - a = 0 := algebra.sub_self
theorem sub_add_cancel : ∀a b : ℤ, a - b + b = a := algebra.sub_add_cancel
theorem add_sub_cancel : ∀a b : ℤ, a + b - b = a := algebra.add_sub_cancel
@ -779,4 +782,11 @@ section port_algebra
@algebra.dvd_of_mul_dvd_mul_right _ _
end port_algebra
/- additional properties -/
theorem of_nat_sub_of_nat {m n : ℕ} (H : #nat m ≥ n) : of_nat m - of_nat n = of_nat (#nat m - n) :=
have H1 : m = (#nat m - n + n), from (nat.sub_add_cancel H)⁻¹,
have H2 : m = (#nat m - n) + n, from congr_arg of_nat H1,
sub_eq_of_eq_add H2
end int

73
library/data/int/order.lean
View file

@ -53,16 +53,17 @@ or.elim (nonneg_or_nonneg_neg (b - a))
have H1 : nonneg (a - b), from H0 ▸ H, -- too bad: can't do it in one step
or.inr H1)
theorem of_nat_le_of_nat (n m : ℕ) : of_nat n ≤ of_nat m ↔ n ≤ m :=
iff.intro
(assume H : of_nat n ≤ of_nat m,
obtain (k : ℕ) (Hk : of_nat n + of_nat k = of_nat m), from le.elim H,
have H2 : n + k = m, from of_nat_inj ((add_of_nat n k)⁻¹ ⬝ Hk),
nat.le.intro H2)
(assume H : n ≤ m,
obtain (k : ℕ) (Hk : n + k = m), from nat.le.elim H,
have H2 : of_nat n + of_nat k = of_nat m, from Hk ▸ add_of_nat n k,
le.intro H2)
theorem of_nat_le_of_nat {m n : ℕ} (H : #nat m ≤ n) : of_nat m ≤ of_nat n :=
obtain (k : ℕ) (Hk : m + k = n), from nat.le.elim H,
le.intro (Hk ▸ of_nat_add_of_nat m k)
theorem le_of_of_nat_le_of_nat {m n : ℕ} (H : of_nat m ≤ of_nat n) : (#nat m ≤ n) :=
obtain (k : ℕ) (Hk : of_nat m + of_nat k = of_nat n), from le.elim H,
have H1 : m + k = n, from of_nat.inj ((of_nat_add_of_nat m k)⁻¹ ⬝ Hk),
nat.le.intro H1
theorem of_nat_le_of_nat_iff (m n : ℕ) : of_nat m ≤ of_nat n ↔ m ≤ n :=
iff.intro le_of_of_nat_le_of_nat of_nat_le_of_nat
theorem lt_add_succ (a : ℤ) (n : ℕ) : a < a + succ n :=
le.intro (show a + 1 + n = a + succ n, from
@ -82,13 +83,19 @@ have H2 : a + succ n = b, from
... = b : Hn,
exists.intro n H2
theorem of_nat_lt_of_nat (n m : ℕ) : of_nat n < of_nat m ↔ n < m :=
theorem of_nat_lt_of_nat_iff (n m : ℕ) : of_nat n < of_nat m ↔ n < m :=
calc
of_nat n < of_nat m ↔ of_nat n + 1 ≤ of_nat m : iff.refl
... ↔ of_nat (succ n) ≤ of_nat m : of_nat_succ n ▸ !iff.refl
... ↔ succ n ≤ m : of_nat_le_of_nat
... ↔ succ n ≤ m : of_nat_le_of_nat_iff
... ↔ n < m : iff.symm (lt_iff_succ_le _ _)
theorem lt_of_of_nat_lt_of_nat {m n : ℕ} (H : of_nat m < of_nat n) : #nat m < n :=
iff.mp !of_nat_lt_of_nat_iff H
theorem of_nat_lt_of_nat {m n : ℕ} (H : #nat m < n) : of_nat m < of_nat n :=
iff.mp' !of_nat_lt_of_nat_iff H
/- show that the integers form an ordered additive group -/
theorem le.refl (a : ℤ) : a ≤ a :=
@ -99,7 +106,7 @@ obtain (n : ℕ) (Hn : a + n = b), from le.elim H1,
obtain (m : ℕ) (Hm : b + m = c), from le.elim H2,
have H3 : a + of_nat (n + m) = c, from
calc
a + of_nat (n + m) = a + (of_nat n + m) : {(add_of_nat n m)⁻¹}
a + of_nat (n + m) = a + (of_nat n + m) : {(of_nat_add_of_nat n m)⁻¹}
... = a + n + m : (add.assoc a n m)⁻¹
... = b + m : {Hn}
... = c : Hm,
@ -110,13 +117,13 @@ obtain (n : ℕ) (Hn : a + n = b), from le.elim H1,
obtain (m : ℕ) (Hm : b + m = a), from le.elim H2,
have H3 : a + of_nat (n + m) = a + 0, from
calc
a + of_nat (n + m) = a + (of_nat n + m) : {(add_of_nat n m)⁻¹}
a + of_nat (n + m) = a + (of_nat n + m) : {(of_nat_add_of_nat n m)⁻¹}
... = a + n + m : (add.assoc a n m)⁻¹
... = b + m : {Hn}
... = a : Hm
... = 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 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
@ -132,7 +139,7 @@ theorem lt.irrefl (a : ℤ) : ¬ a < a :=
a + succ n = a : Hn
... = a + 0 : by simp,
have H3 : succ n = 0, from add.left_cancel H2,
have H4 : succ n = 0, from of_nat_inj H3,
have H4 : succ n = 0, from of_nat.inj H3,
absurd H4 !succ_ne_zero)
theorem ne_of_lt {a b : ℤ} (H : a < b) : a ≠ b :=
@ -205,7 +212,7 @@ lt.intro
... = succ n * b : nat.zero_add
... = succ n * (0 + succ m) : {Hm⁻¹}
... = succ n * succ m : nat.zero_add
... = of_nat (succ n * succ m) : mul_of_nat
... = of_nat (succ n * succ m) : of_nat_mul_of_nat
... = of_nat (succ n * m + succ n) : nat.mul_succ
... = of_nat (succ (succ n * m + n)) : nat.add_succ
... = 0 + succ (succ n * m + n) : zero_add))
@ -443,6 +450,8 @@ section port_algebra
@algebra.sub_lt_sub_of_le_of_lt _ _
theorem sub_lt_sub_of_lt_of_le : ∀{a b c d : ℤ}, a < b → c ≤ d → a - d < b - c :=
@algebra.sub_lt_sub_of_lt_of_le _ _
theorem sub_le_self : ∀(a : ℤ) {b : ℤ}, b ≥ 0 → a - b ≤ a := algebra.sub_le_self
theorem sub_lt_self : ∀(a : ℤ) {b : ℤ}, b > 0 → a - b < a := algebra.sub_lt_self
theorem eq_zero_of_neg_eq : ∀{a : ℤ}, -a = a → a = 0 := @algebra.eq_zero_of_neg_eq _ _
definition abs : ℤ → ℤ := algebra.abs
@ -461,7 +470,7 @@ section port_algebra
theorem abs_eq_zero_iff_eq_zero : ∀a : ℤ, |a| = 0 ↔ a = 0 := algebra.abs_eq_zero_iff_eq_zero
theorem abs_pos_of_pos : ∀{a : ℤ}, a > 0 → |a| > 0 := @algebra.abs_pos_of_pos _ _
theorem abs_pos_of_neg : ∀{a : ℤ}, a < 0 → |a| > 0 := @algebra.abs_pos_of_neg _ _
theorem abs_pos_of_ne_zero : ∀a : ℤ, a ≠ 0 → |a| > 0 := @algebra.abs_pos_of_ne_zero _ _
theorem abs_pos_of_ne_zero : ∀{a : ℤ}, a ≠ 0 → |a| > 0 := @algebra.abs_pos_of_ne_zero _ _
theorem abs_sub : ∀a b : ℤ, |a - b| = |b - a| := algebra.abs_sub
theorem abs.by_cases : ∀{P : ℤ → Prop}, ∀{a : ℤ}, P a → P (-a) → P (|a|) :=
@algebra.abs.by_cases _ _
@ -571,4 +580,32 @@ calc
of_nat (nat_abs a) = of_nat (nat_abs (-a)) : nat_abs_neg
... = -a : of_nat_nat_abs_of_nonneg H1
theorem of_nat_nat_abs (b : ℤ) : nat_abs b = |b| :=
or.elim (le.total 0 b)
(assume H : b ≥ 0, of_nat_nat_abs_of_nonneg H ⬝ (abs_of_nonneg H)⁻¹)
(assume H : b ≤ 0, of_nat_nat_abs_of_nonpos H ⬝ (abs_of_nonpos H)⁻¹)
theorem lt_of_add_one_le {a b : ℤ} (H : a + 1 ≤ b) : a < b :=
obtain n (H1 : a + 1 + n = b), from le.elim H,
have H2 : a + succ n = b, by rewrite [-H1, add.assoc, (add.comm 1)],
lt.intro H2
theorem add_one_le_of_lt {a b : ℤ} (H : a < b) : a + 1 ≤ b :=
obtain n (H1 : a + succ n = b), from lt.elim H,
have H2 : a + 1 + n = b, by rewrite [-H1, add.assoc, (add.comm 1)],
le.intro H2
theorem of_nat_nonneg (n : ℕ) : of_nat n ≥ 0 := trivial
theorem of_nat_pos {n : ℕ} (Hpos : #nat n > 0) : of_nat n > 0 :=
of_nat_lt_of_nat Hpos
theorem sign_of_succ (n : nat) : sign (succ n) = 1 :=
sign_of_pos (of_nat_pos !nat.succ_pos)
theorem exists_eq_neg_succ_of_nat {a : ℤ} : a < 0 → ∃m : ℕ, a = -[m +1] :=
int.cases_on a
(take m H, absurd (of_nat_nonneg m) (not_le_of_lt H))
(take m H, exists.intro m rfl)
end int