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refactor(library/hott): remove more unnecessary annotations

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This commit is contained in:
Leonardo de Moura 2016-02-25 14:30:00 -08:00
parent 510168a387
commit 768ba1c363
36 changed files with 63 additions and 83 deletions
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3
hott/algebra/field.hlean
View file

@ -334,8 +334,7 @@ section discrete_field
(suppose x ≠ 0,
sum.inr (by rewrite [-one_mul, -(inv_mul_cancel this), mul.assoc, H, mul_zero]))
definition discrete_field.to_integral_domain [trans_instance] [reducible] :
integral_domain A :=
definition discrete_field.to_integral_domain [trans_instance] : integral_domain A :=
⦃ integral_domain, s,
eq_zero_sum_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_sum_eq_zero_of_mul_eq_zero⦄

12
hott/algebra/group.hlean
View file

@ -316,13 +316,11 @@ section group
... = x ∘c b : Py
... = a : Px)
definition group.to_left_cancel_semigroup [trans_instance] [reducible] :
left_cancel_semigroup A :=
definition group.to_left_cancel_semigroup [trans_instance] : left_cancel_semigroup A :=
⦃ left_cancel_semigroup, s,
mul_left_cancel := @mul_left_cancel A s ⦄
definition group.to_right_cancel_semigroup [trans_instance] [reducible] :
right_cancel_semigroup A :=
definition group.to_right_cancel_semigroup [trans_instance] : right_cancel_semigroup A :=
⦃ right_cancel_semigroup, s,
mul_right_cancel := @mul_right_cancel A s ⦄
@ -444,13 +442,11 @@ section add_group
... = (c + b) + -b : H
... = c : add_neg_cancel_right
definition add_group.to_left_cancel_semigroup [trans_instance] [reducible] :
add_left_cancel_semigroup A :=
definition add_group.to_left_cancel_semigroup [trans_instance] : add_left_cancel_semigroup A :=
⦃ add_left_cancel_semigroup, s,
add_left_cancel := @add_left_cancel A s ⦄
definition add_group.to_add_right_cancel_semigroup [trans_instance] [reducible] :
add_right_cancel_semigroup A :=
definition add_group.to_add_right_cancel_semigroup [trans_instance] : add_right_cancel_semigroup A :=
⦃ add_right_cancel_semigroup, s,
add_right_cancel := @add_right_cancel A s ⦄

7
hott/algebra/order.hlean
View file

@ -109,7 +109,7 @@ section
private theorem lt_trans (s' : order_pair A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
lt_of_lt_of_le lt_ab (le_of_lt lt_bc)
definition order_pair.to_strict_order [trans_instance] [reducible] : strict_order A :=
definition order_pair.to_strict_order [trans_instance] : strict_order A :=
⦃ strict_order, s, lt_irrefl := lt_irrefl s, lt_trans := lt_trans s ⦄
definition gt_of_gt_of_ge [trans] (H1 : a > b) (H2 : b ≥ c) : a > c := lt_of_le_of_lt H2 H1
@ -179,8 +179,7 @@ have ne_ac : a ≠ c, from
show empty, from ne_of_lt' lt_bc eq_bc,
show a < c, from iff.mpr (lt_iff_le_prod_ne) (pair le_ac ne_ac)
definition strong_order_pair.to_order_pair [trans_instance] [reducible]
[s : strong_order_pair A] : order_pair A :=
definition strong_order_pair.to_order_pair [trans_instance] [s : strong_order_pair A] : order_pair A :=
⦃ order_pair, s,
lt_irrefl := lt_irrefl',
le_of_lt := le_of_lt',
@ -194,7 +193,7 @@ structure linear_order_pair [class] (A : Type) extends order_pair A, linear_weak
structure linear_strong_order_pair [class] (A : Type) extends strong_order_pair A,
linear_weak_order A
definition linear_strong_order_pair.to_linear_order_pair [trans_instance] [reducible]
definition linear_strong_order_pair.to_linear_order_pair [trans_instance]
[s : linear_strong_order_pair A] : linear_order_pair A :=
⦃ linear_order_pair, s, strong_order_pair.to_order_pair ⦄

3
hott/algebra/ordered_field.hlean
View file

@ -384,8 +384,7 @@ section discrete_linear_ordered_field
(assume H' : ¬ y < x,
decidable.inl (le.antisymm (le_of_not_gt H') (le_of_not_gt H))))
definition discrete_linear_ordered_field.to_discrete_field [trans_instance] [reducible]
: discrete_field A :=
definition discrete_linear_ordered_field.to_discrete_field [trans_instance] : discrete_field A :=
⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
theorem pos_of_one_div_pos (H : 0 < 1 / a) : 0 < a :=

4
hott/algebra/ordered_group.hlean
View file

@ -209,7 +209,7 @@ theorem ordered_comm_group.lt_of_add_lt_add_left [s : ordered_comm_group A] {a b
assert H' : -a + (a + b) < -a + (a + c), from ordered_comm_group.add_lt_add_left _ _ H _,
by rewrite *neg_add_cancel_left at H'; exact H'
definition ordered_comm_group.to_ordered_cancel_comm_monoid [trans_instance] [reducible]
definition ordered_comm_group.to_ordered_cancel_comm_monoid [trans_instance]
[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
⦃ ordered_cancel_comm_monoid, s,
add_left_cancel := @add.left_cancel A _,
@ -581,7 +581,7 @@ structure decidable_linear_ordered_comm_group [class] (A : Type)
(add_lt_add_left : Π a b, lt a b → Π c, lt (add c a) (add c b))
definition decidable_linear_ordered_comm_group.to_ordered_comm_group
[trans_instance] [reducible]
[trans_instance]
(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
⦃ ordered_comm_group, s,
le_of_lt := @le_of_lt A _,

10
hott/algebra/ordered_ring.hlean
View file

@ -234,9 +234,7 @@ begin
exact (iff.mp !sub_pos_iff_lt H2)
end
definition ordered_ring.to_ordered_semiring [trans_instance] [reducible]
[s : ordered_ring A] :
ordered_semiring A :=
definition ordered_ring.to_ordered_semiring [trans_instance] [s : ordered_ring A] : ordered_semiring A :=
⦃ ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
@ -316,9 +314,7 @@ structure linear_ordered_ring [class] (A : Type)
extends ordered_ring A, linear_strong_order_pair A :=
(zero_lt_one : lt zero one)
definition linear_ordered_ring.to_linear_ordered_semiring [trans_instance] [reducible]
[s : linear_ordered_ring A] :
linear_ordered_semiring A :=
definition linear_ordered_ring.to_linear_ordered_semiring [trans_instance] [s : linear_ordered_ring A] : linear_ordered_semiring A :=
⦃ linear_ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
@ -370,7 +366,7 @@ lt.by_cases
end))
-- Linearity implies no zero divisors. Doesn't need commutativity.
definition linear_ordered_comm_ring.to_integral_domain [trans_instance] [reducible]
definition linear_ordered_comm_ring.to_integral_domain [trans_instance]
[s: linear_ordered_comm_ring A] : integral_domain A :=
⦃ integral_domain, s,
eq_zero_sum_eq_zero_of_mul_eq_zero :=

4
hott/algebra/ring.hlean
View file

@ -172,7 +172,7 @@ have 0 * a + 0 = 0 * a + 0 * a, from calc
... = 0 * a + 0 * a : by rewrite {_*a}ring.right_distrib,
show 0 * a = 0, from (add.left_cancel this)⁻¹
definition ring.to_semiring [trans_instance] [reducible] [s : ring A] : semiring A :=
definition ring.to_semiring [trans_instance] [s : ring A] : semiring A :=
⦃ semiring, s,
mul_zero := ring.mul_zero,
zero_mul := ring.zero_mul ⦄
@ -260,7 +260,7 @@ end
structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
definition comm_ring.to_comm_semiring [trans_instance] [reducible] [s : comm_ring A] : comm_semiring A :=
definition comm_ring.to_comm_semiring [trans_instance] [s : comm_ring A] : comm_semiring A :=
⦃ comm_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul ⦄

2
hott/types/int/basic.hlean
View file

@ -553,7 +553,7 @@ protected theorem eq_zero_sum_eq_zero_of_mul_eq_zero {a b : ℤ} (H : a * b = 0)
sum.imp eq_zero_of_nat_abs_eq_zero eq_zero_of_nat_abs_eq_zero
(eq_zero_sum_eq_zero_of_mul_eq_zero (by rewrite [-nat_abs_mul, H]))
protected definition integral_domain [reducible] [trans_instance] : integral_domain int :=
protected definition integral_domain [trans_instance] : integral_domain int :=
⦃integral_domain,
add := int.add,
add_assoc := int.add_assoc,

2
hott/types/nat/basic.hlean
View file

@ -287,7 +287,7 @@ nat.cases_on n
... = succ (succ n' * m' + n') : add_succ)⁻¹
!succ_ne_zero))
protected definition comm_semiring [reducible] [trans_instance] : comm_semiring nat :=
protected definition comm_semiring [trans_instance] : comm_semiring nat :=
⦃comm_semiring,
add := nat.add,
add_assoc := nat.add_assoc,

2
hott/types/nat/order.hlean
View file

@ -90,7 +90,7 @@ protected theorem mul_lt_mul_of_pos_right {n m k : ℕ} (H : n < m) (Hk : k > 0)
/- nat is an instance of a linearly ordered semiring and a lattice -/
protected definition decidable_linear_ordered_semiring [reducible] [trans_instance] :
protected definition decidable_linear_ordered_semiring [trans_instance] :
decidable_linear_ordered_semiring nat :=
⦃ decidable_linear_ordered_semiring, nat.comm_semiring,
add_left_cancel := @nat.add_left_cancel,

4
library/algebra/complete_lattice.lean
View file

@ -113,7 +113,7 @@ definition complete_lattice_Inf_to_complete_lattice_Sup [C : complete_lattice_In
⦃ complete_lattice_Sup, C ⦄
-- Every complete_lattice_Inf is a complete_lattice
definition complete_lattice_Inf_to_complete_lattice [trans_instance] [reducible] [C : complete_lattice_Inf A] :
definition complete_lattice_Inf_to_complete_lattice [trans_instance] [C : complete_lattice_Inf A] :
complete_lattice A :=
⦃ complete_lattice, C ⦄
@ -179,7 +179,7 @@ definition complete_lattice_Sup_to_complete_lattice_Inf [C : complete_lattice_Su
-- Every complete_lattice_Sup is a complete_lattice
section
definition complete_lattice_Sup_to_complete_lattice [trans_instance] [reducible] [C : complete_lattice_Sup A] :
definition complete_lattice_Sup_to_complete_lattice [trans_instance] [C : complete_lattice_Sup A] :
complete_lattice A :=
⦃ complete_lattice, C ⦄
end

2
library/algebra/field.lean
View file

@ -313,7 +313,7 @@ section discrete_field
(suppose x ≠ 0,
or.inr (by rewrite [-one_mul, -(inv_mul_cancel this), mul.assoc, H, mul_zero]))
definition discrete_field.to_integral_domain [trans_instance] [reducible] :
definition discrete_field.to_integral_domain [trans_instance] :
integral_domain A :=
⦃ integral_domain, s,
eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero⦄

10
library/algebra/group.lean
View file

@ -288,12 +288,12 @@ section group
end group
definition group.to_left_cancel_semigroup [trans_instance] [reducible] [s : group A] :
definition group.to_left_cancel_semigroup [trans_instance] [s : group A] :
left_cancel_semigroup A :=
⦃ left_cancel_semigroup, s,
mul_left_cancel := @mul_left_cancel A s ⦄
definition group.to_right_cancel_semigroup [trans_instance] [reducible] [s : group A] :
definition group.to_right_cancel_semigroup [trans_instance] [s : group A] :
right_cancel_semigroup A :=
⦃ right_cancel_semigroup, s,
mul_right_cancel := @mul_right_cancel A s ⦄
@ -410,13 +410,11 @@ section add_group
assert a + b + -b = a, by inst_simp,
by inst_simp
definition add_group.to_left_cancel_semigroup [trans_instance] [reducible] :
add_left_cancel_semigroup A :=
definition add_group.to_left_cancel_semigroup [trans_instance] : add_left_cancel_semigroup A :=
⦃ add_left_cancel_semigroup, s,
add_left_cancel := @add_left_cancel A s ⦄
definition add_group.to_add_right_cancel_semigroup [trans_instance] [reducible] :
add_right_cancel_semigroup A :=
definition add_group.to_add_right_cancel_semigroup [trans_instance] : add_right_cancel_semigroup A :=
⦃ add_right_cancel_semigroup, s,
add_right_cancel := @add_right_cancel A s ⦄

6
library/algebra/order.lean
View file

@ -111,7 +111,7 @@ section
private theorem lt_trans (s' : order_pair A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
lt_of_lt_of_le lt_ab (le_of_lt lt_bc)
definition order_pair.to_strict_order [trans_instance] [reducible] : strict_order A :=
definition order_pair.to_strict_order [trans_instance] : strict_order A :=
⦃ strict_order, s, lt_irrefl := lt_irrefl s, lt_trans := lt_trans s ⦄
theorem gt_of_gt_of_ge [trans] (H1 : a > b) (H2 : b ≥ c) : a > c := lt_of_le_of_lt H2 H1
@ -181,7 +181,7 @@ have ne_ac : a ≠ c, from
show false, from ne_of_lt' lt_bc eq_bc,
show a < c, from iff.mpr (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
definition strong_order_pair.to_order_pair [trans_instance] [reducible]
definition strong_order_pair.to_order_pair [trans_instance]
[s : strong_order_pair A] : order_pair A :=
⦃ order_pair, s,
lt_irrefl := lt_irrefl',
@ -196,7 +196,7 @@ structure linear_order_pair [class] (A : Type) extends order_pair A, linear_weak
structure linear_strong_order_pair [class] (A : Type) extends strong_order_pair A,
linear_weak_order A
definition linear_strong_order_pair.to_linear_order_pair [trans_instance] [reducible]
definition linear_strong_order_pair.to_linear_order_pair [trans_instance]
[s : linear_strong_order_pair A] : linear_order_pair A :=
⦃ linear_order_pair, s, strong_order_pair.to_order_pair ⦄

3
library/algebra/ordered_field.lean
View file

@ -393,8 +393,7 @@ section discrete_linear_ordered_field
(assume H' : ¬ y < x,
decidable.inl (le.antisymm (le_of_not_gt H') (le_of_not_gt H))))
definition discrete_linear_ordered_field.to_discrete_field [trans_instance] [reducible]
: discrete_field A :=
definition discrete_linear_ordered_field.to_discrete_field [trans_instance] : discrete_field A :=
⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
theorem pos_of_one_div_pos (H : 0 < 1 / a) : 0 < a :=

7
library/algebra/ordered_group.lean
View file

@ -249,8 +249,7 @@ theorem ordered_comm_group.lt_of_add_lt_add_left [ordered_comm_group A] {a b c :
assert H' : -a + (a + b) < -a + (a + c), from ordered_comm_group.add_lt_add_left _ _ H _,
by rewrite *neg_add_cancel_left at H'; exact H'
definition ordered_comm_group.to_ordered_cancel_comm_monoid [trans_instance] [reducible]
[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
definition ordered_comm_group.to_ordered_cancel_comm_monoid [trans_instance] [s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
⦃ ordered_cancel_comm_monoid, s,
add_left_cancel := @add.left_cancel A _,
add_right_cancel := @add.right_cancel A _,
@ -621,7 +620,7 @@ structure decidable_linear_ordered_comm_group [class] (A : Type)
(add_lt_add_left : ∀ a b, lt a b → ∀ c, lt (add c a) (add c b))
definition decidable_linear_ordered_comm_group.to_ordered_comm_group
[trans_instance] [reducible]
[trans_instance]
(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
⦃ ordered_comm_group, s,
le_of_lt := @le_of_lt A _,
@ -629,7 +628,7 @@ definition decidable_linear_ordered_comm_group.to_ordered_comm_group
lt_of_lt_of_le := @lt_of_lt_of_le A _ ⦄
definition decidable_linear_ordered_comm_group.to_decidable_linear_ordered_cancel_comm_monoid
[trans_instance] [reducible] (A : Type) [s : decidable_linear_ordered_comm_group A] :
[trans_instance] (A : Type) [s : decidable_linear_ordered_comm_group A] :
decidable_linear_ordered_cancel_comm_monoid A :=
⦃ decidable_linear_ordered_cancel_comm_monoid, s,
@ordered_comm_group.to_ordered_cancel_comm_monoid A _ ⦄

8
library/algebra/ordered_ring.lean
View file

@ -242,7 +242,7 @@ begin
exact (iff.mp !sub_pos_iff_lt H2)
end
definition ordered_ring.to_ordered_semiring [trans_instance] [reducible]
definition ordered_ring.to_ordered_semiring [trans_instance]
[s : ordered_ring A] :
ordered_semiring A :=
⦃ ordered_semiring, s,
@ -324,7 +324,7 @@ structure linear_ordered_ring [class] (A : Type)
extends ordered_ring A, linear_strong_order_pair A :=
(zero_lt_one : lt zero one)
definition linear_ordered_ring.to_linear_ordered_semiring [trans_instance] [reducible]
definition linear_ordered_ring.to_linear_ordered_semiring [trans_instance]
[s : linear_ordered_ring A] :
linear_ordered_semiring A :=
⦃ linear_ordered_semiring, s,
@ -378,7 +378,7 @@ lt.by_cases
end))
-- Linearity implies no zero divisors. Doesn't need commutativity.
definition linear_ordered_comm_ring.to_integral_domain [trans_instance] [reducible]
definition linear_ordered_comm_ring.to_integral_domain [trans_instance]
[s: linear_ordered_comm_ring A] : integral_domain A :=
⦃ integral_domain, s,
eq_zero_or_eq_zero_of_mul_eq_zero :=
@ -464,7 +464,7 @@ structure decidable_linear_ordered_comm_ring [class] (A : Type) extends linear_o
decidable_linear_ordered_comm_group A
definition decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_semiring
[trans_instance] [reducible] [s : decidable_linear_ordered_comm_ring A] :
[trans_instance] [s : decidable_linear_ordered_comm_ring A] :
decidable_linear_ordered_semiring A :=
⦃decidable_linear_ordered_semiring, s, @linear_ordered_ring.to_linear_ordered_semiring A _⦄

4
library/algebra/ring.lean
View file

@ -174,7 +174,7 @@ have 0 * a + 0 = 0 * a + 0 * a, from calc
... = 0 * a + 0 * a : by rewrite right_distrib,
show 0 * a = 0, from (add.left_cancel this)⁻¹
definition ring.to_semiring [trans_instance] [reducible] [s : ring A] : semiring A :=
definition ring.to_semiring [trans_instance] [s : ring A] : semiring A :=
⦃ semiring, s,
mul_zero := ring.mul_zero,
zero_mul := ring.zero_mul ⦄
@ -255,7 +255,7 @@ end
structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
definition comm_ring.to_comm_semiring [trans_instance] [reducible] [s : comm_ring A] : comm_semiring A :=
definition comm_ring.to_comm_semiring [trans_instance] [s : comm_ring A] : comm_semiring A :=
⦃ comm_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul ⦄

2
library/data/complex.lean
View file

@ -278,7 +278,7 @@ take z w, classical.prop_decidable (z = w)
protected theorem zero_ne_one : (0 : ℂ) ≠ 1 :=
assume H, zero_ne_one (eq_of_of_real_eq_of_real H)
protected noncomputable definition discrete_field [reducible][trans_instance] :
protected noncomputable definition discrete_field [trans_instance] :
discrete_field ℂ :=
⦃ discrete_field, complex.comm_ring,
mul_inv_cancel := @complex.mul_inv_cancel,

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

@ -515,7 +515,7 @@ protected theorem eq_zero_or_eq_zero_of_mul_eq_zero {a b : ℤ} (H : a * b = 0)
or.imp eq_zero_of_nat_abs_eq_zero eq_zero_of_nat_abs_eq_zero
(eq_zero_or_eq_zero_of_mul_eq_zero (by rewrite [-nat_abs_mul, H]))
protected definition integral_domain [reducible] [trans_instance] : integral_domain int :=
protected definition integral_domain [trans_instance] : integral_domain int :=
⦃integral_domain,
add := int.add,
add_assoc := int.add_assoc,

2
library/data/int/order.lean
View file

@ -232,7 +232,7 @@ protected theorem lt_of_le_of_lt {a b c : ℤ} (Hab : a ≤ b) (Hbc : b < c) :
(iff.mpr !int.lt_iff_le_and_ne) (and.intro Hac
(assume Heq, int.not_le_of_gt (Heq⁻¹ ▸ Hbc) Hab))
protected definition linear_ordered_comm_ring [reducible] [trans_instance] :
protected definition linear_ordered_comm_ring [trans_instance] :
linear_ordered_comm_ring int :=
⦃linear_ordered_comm_ring, int.integral_domain,
le := int.le,

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

@ -237,7 +237,7 @@ nat.cases_on n (by simp)
(show succ (succ n' * m' + n') = 0, by simp)
!succ_ne_zero))
protected definition comm_semiring [reducible] [trans_instance] : comm_semiring nat :=
protected definition comm_semiring [trans_instance] : comm_semiring nat :=
⦃comm_semiring,
add := nat.add,
add_assoc := nat.add_assoc,

2
library/data/nat/order.lean
View file

@ -90,7 +90,7 @@ protected theorem mul_lt_mul_of_pos_right {n m k : ℕ} (H : n < m) (Hk : k > 0)
/- nat is an instance of a linearly ordered semiring and a lattice -/
protected definition decidable_linear_ordered_semiring [reducible] [trans_instance] :
protected definition decidable_linear_ordered_semiring [trans_instance] :
decidable_linear_ordered_semiring nat :=
⦃ decidable_linear_ordered_semiring, nat.comm_semiring,
add_left_cancel := @nat.add_left_cancel,

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

@ -550,7 +550,7 @@ decidable.by_cases
end))
protected definition discrete_field [reducible] [trans_instance] : discrete_field rat :=
protected definition discrete_field [trans_instance] : discrete_field rat :=
⦃discrete_field,
add := rat.add,
add_assoc := rat.add_assoc,

2
library/data/rat/order.lean
View file

@ -301,7 +301,7 @@ let H' := rat.le_of_lt H in
(take Heq, let Heq' := add_left_cancel Heq in
!rat.lt_irrefl (Heq' ▸ H)))
protected definition discrete_linear_ordered_field [reducible] [trans_instance] :
protected definition discrete_linear_ordered_field [trans_instance] :
discrete_linear_ordered_field rat :=
⦃discrete_linear_ordered_field,
rat.discrete_field,

2
library/data/real/division.lean
View file

@ -639,7 +639,7 @@ noncomputable definition dec_lt : decidable_rel real.lt :=
apply prop_decidable
end
protected noncomputable definition discrete_linear_ordered_field [reducible] [trans_instance]:
protected noncomputable definition discrete_linear_ordered_field [trans_instance]:
discrete_linear_ordered_field ℝ :=
⦃ discrete_linear_ordered_field, real.comm_ring, real.ordered_ring,
le_total := real.le_total,

2
library/data/real/order.lean
View file

@ -1092,7 +1092,7 @@ protected theorem le_of_lt_or_eq (x y : ℝ) : x < y ∨ x = y → x ≤ y :=
apply (or.inr (quot.exact H'))
end)))
definition ordered_ring [reducible] [trans_instance] : ordered_ring ℝ :=
definition ordered_ring [trans_instance] : ordered_ring ℝ :=
⦃ ordered_ring, real.comm_ring,
le_refl := real.le_refl,
le_trans := @real.le_trans,

2
library/data/set/filter.lean
View file

@ -287,7 +287,7 @@ namespace complete_lattice
Sup_le (λ G GS, GS F FS)
end complete_lattice
protected definition complete_lattice [reducible] [trans_instance] : complete_lattice (filter A) :=
protected definition complete_lattice [trans_instance] : complete_lattice (filter A) :=
⦃ complete_lattice,
le := complete_lattice.le,
le_refl := complete_lattice.le_refl,

2
library/theories/analysis/complex_norm.lean
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@ -63,7 +63,7 @@ namespace complex
local attribute complex.real_inner_product_space [trans_instance]
protected definition normed_vector_space [trans_instance] [reducible] : normed_vector_space ℂ :=
protected definition normed_vector_space [trans_instance] : normed_vector_space ℂ :=
_
theorem norm_squared_eq_cmod (z : ℂ) : ∥ z ∥^2 = cmod z := by rewrite norm_squared

2
library/theories/analysis/inner_product.lean
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@ -172,7 +172,7 @@ have (ip_norm (u + v))^2 ≤ (ip_norm u + ip_norm v)^2, from
mul.comm (ip_norm v) (ip_norm u)],
le_of_squared_le_squared (add_nonneg !ip_norm_nonneg !ip_norm_nonneg) this
definition inner_product_space.to_normed_space [trans_instance] [reducible] :
definition inner_product_space.to_normed_space [trans_instance] :
normed_vector_space V :=
⦃ normed_vector_space, _inst_1,
norm := ip_norm,

5
library/theories/analysis/normed_space.lean
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@ -103,7 +103,7 @@ section
private lemma nvs_dist_comm (u v : V) : nvs_dist u v = nvs_dist v u :=
by rewrite [↑nvs_dist, -norm_neg, neg_sub]
definition normed_vector_space_to_metric_space [reducible] [trans_instance]
definition normed_vector_space_to_metric_space [trans_instance]
(V : Type) [nvsV : normed_vector_space V] :
metric_space V :=
⦃ metric_space,
@ -132,8 +132,7 @@ structure banach_space [class] (V : Type) extends nvsV : normed_vector_space V :
(complete : ∀ X, @analysis.cauchy V (@normed_vector_space_to_metric_space V nvsV) X →
@analysis.converges_seq V (@normed_vector_space_to_metric_space V nvsV) X)
definition banach_space_to_metric_space [reducible] [trans_instance]
(V : Type) [bsV : banach_space V] :
definition banach_space_to_metric_space [trans_instance] (V : Type) [bsV : banach_space V] :
complete_metric_space V :=
⦃ complete_metric_space, normed_vector_space_to_metric_space V,
complete := banach_space.complete

4
library/theories/analysis/real_limit.lean
View file

@ -221,7 +221,7 @@ exists.intro l
end analysis
definition complete_metric_space_real [reducible] [trans_instance] :
definition complete_metric_space_real [trans_instance] :
complete_metric_space ℝ :=
⦃complete_metric_space, metric_space_real,
complete := @analysis.converges_seq_of_cauchy
@ -238,7 +238,7 @@ definition real_vector_space_real : real_vector_space ℝ :=
one_smul := one_mul
⦄
definition banach_space_real [trans_instance] [reducible] : banach_space ℝ :=
definition banach_space_real [trans_instance] : banach_space ℝ :=
⦃ banach_space, real_vector_space_real,
norm := abs,
norm_zero := abs_zero,

12
library/theories/measure_theory/extended_real.lean
View file

@ -281,8 +281,7 @@ by krewrite [ereal.mul_comm, ereal.one_mul]
-- Note that distributivity fails, e.g. ∞ ⬝ (-1 + 1) ≠ ∞ * -1 + ∞ * 1
protected definition comm_monoid [reducible] [trans_instance] :
comm_monoid ereal :=
protected definition comm_monoid [trans_instance] : comm_monoid ereal :=
⦃comm_monoid,
mul := ereal.mul,
mul_assoc := ereal.mul_assoc,
@ -292,8 +291,7 @@ protected definition comm_monoid [reducible] [trans_instance] :
mul_comm := ereal.mul_comm
⦄
protected definition add_comm_monoid [reducible] [trans_instance] :
add_comm_monoid ereal :=
protected definition add_comm_monoid [trans_instance] : add_comm_monoid ereal :=
⦃add_comm_monoid,
add := ereal.add,
add_assoc := ereal.add_assoc,
@ -313,8 +311,7 @@ protected definition le : ereal → ereal → Prop
| ∞ (of_real y) := false
| ∞ -∞ := false
definition ereal_has_le [instance] [priority ereal.prio] :
has_le ereal :=
definition ereal_has_le [instance] [priority ereal.prio] : has_le ereal :=
has_le.mk ereal.le
theorem of_real_le_of_real (x y : real) : of_real x ≤ of_real y ↔ x ≤ y :=
@ -393,8 +390,7 @@ theorem neg_infty_lt_of_real (x : real) : -∞ < of_real x := and.intro trivial
theorem of_real_lt_infty (x : real) : of_real x < ∞ := and.intro trivial (ne.symm !infty_ne_of_real)
protected definition decidable_linear_order [reducible] [trans_instance] :
decidable_linear_order ereal :=
protected definition decidable_linear_order [trans_instance] : decidable_linear_order ereal :=
⦃decidable_linear_order,
le := ereal.le,
le_refl := ereal.le_refl,

2
library/theories/measure_theory/sigma_algebra.lean
View file

@ -200,7 +200,7 @@ protected theorem Sup_le {N : sigma_algebra X} {MS : set (sigma_algebra X)} (H :
have (⋃ M ∈ MS, @sets _ M) ⊆ @sets _ N, from bUnion_subset H,
sets_generated_by_initial this
protected definition complete_lattice [reducible] [trans_instance] :
protected definition complete_lattice [trans_instance] :
complete_lattice (sigma_algebra X) :=
⦃complete_lattice,
le := sigma_algebra.le,

4
library/theories/topology/basic.lean
View file

@ -166,7 +166,7 @@ end topology
structure T1_space [class] (X : Type) extends topology X :=
(T1 : ∀ {x y}, x ≠ y → ∃ U, U ∈ opens ∧ x ∈ U ∧ y ∉ U)
protected definition T0_space.of_T1 [reducible] [trans_instance] {X : Type} [T : T1_space X] :
protected definition T0_space.of_T1 [trans_instance] {X : Type} [T : T1_space X] :
T0_space X :=
⦃T0_space, T,
T0 := abstract
@ -208,7 +208,7 @@ end topology
structure T2_space [class] (X : Type) extends topology X :=
(T2 : ∀ {x y}, x ≠ y → ∃ U V, U ∈ opens ∧ V ∈ opens ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = ∅)
protected definition T1_space.of_T2 [reducible] [trans_instance] {X : Type} [T : T2_space X] :
protected definition T1_space.of_T2 [trans_instance] {X : Type} [T : T2_space X] :
T1_space X :=
⦃T1_space, T,
T1 := abstract

2
library/theories/topology/order_topology.lean
View file

@ -57,7 +57,7 @@ section
exists.intro y (exists.intro x (and.intro (and.intro `x < y` `x < y`) this))
end
protected definition T2_space.of_linorder_topology [reducible] [trans_instance] :
protected definition T2_space.of_linorder_topology [trans_instance] :
T2_space X :=
⦃ T2_space, linorder_topology,
T2 := abstract