core / init.core - mathlib docs

def id_delta {α : Sort u} (a : α) :

α

The kernel definitional equality test (t =?= s) has special support for id_delta applications. It implements the following rules

(id_delta t) =?= t
t =?= (id_delta t)
(id_delta t) =?= s IF (unfold_of t) =?= s
t =?= id_delta s IF t =?= (unfold_of s)

This is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.

We use id_delta applications to address performance problems when type checking lemmas generated by the equation compiler.

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@[reducible]

def opt_param (α : Sort u) (default : α) :

Sort u

Gadget for optional parameter support.

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@[reducible]

def out_param (α : Sort u) :

Sort u

Gadget for marking output parameters in type classes.

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@[reducible]

def id_rhs (α : Sort u) (a : α) :

α

id_rhs is an auxiliary declaration used in the equation compiler to address performance issues when proving equational lemmas. The equation compiler uses it as a marker.

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structure punit :

Sort u

Instances for punit

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@[reducible]

def unit :

Type

An abbreviation for punit.{0}, its most common instantiation. This type should be preferred over punit where possible to avoid unnecessary universe parameters.

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@[reducible]

def unit.star :

unit

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@[reducible]

def thunk (α : Type u) :

Type u

Gadget for defining thunks, thunk parameters have special treatment. Example: given def f (s : string) (t : thunk nat) : nat an application f "hello" 10 is converted into f "hello" (λ _, 10)

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structure true :

Prop

Instances for true

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inductive false :

Prop

Instances for false

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inductive empty :

Type

Instances for empty

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def not (a : Prop) :

Prop

Logical not.

not P, with notation ¬ P, is the Prop which is true if and only if P is false. It is internally represented as P → false, so one way to prove a goal ⊢ ¬ P is to use intro h, which gives you a new hypothesis h : P and the goal ⊢ false.

A hypothesis h : ¬ P can be used in term mode as a function, so if w : P then h w : false.

Related mathlib tactic: contrapose.

Instances for not

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inductive eq {α : Sort u} (a : α) :

α → Prop

refl : ∀ {α : Sort u} (a : α), a = a

Instances for eq

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inductive heq {α : Sort u} (a : α) {β : Sort u} :

β → Prop

refl : ∀ {α : Sort u} (a : α), a == a

Heterogeneous equality.

Its purpose is to write down equalities between terms whose types are not definitionally equal. For example, given x : vector α n and y : vector α (0+n), x = y doesn't typecheck but x == y does.

If you have a goal ⊢ x == y, your first instinct should be to ask (either yourself, or on zulip) if something has gone wrong already. If you really do need to follow this route, you may find the lemmas eq_rec_heq and eq_mpr_heq useful.

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structure prod (α : Type u) (β : Type v) :

Type (max u v)

fst : α
snd : β

Instances for prod

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structure pprod (α : Sort u) (β : Sort v) :

Sort (max 1 u v)

fst : α
snd : β

Similar to prod, but α and β can be propositions. We use this type internally to automatically generate the brec_on recursor.

Instances for pprod

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structure and (a b : Prop) :

Prop

left : a
right : b

Logical and.

and P Q, with notation P ∧ Q, is the Prop which is true precisely when P and Q are both true.

To prove a goal ⊢ P ∧ Q, you can use the tactic split, which gives two separate goals ⊢ P and ⊢ Q.

Given a hypothesis h : P ∧ Q, you can use the tactic cases h with hP hQ to obtain two new hypotheses hP : P and hQ : Q. See also the obtain or rcases tactics in mathlib.

Instances for and

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theorem and.elim_left {a b : Prop} (h : a ∧ b) :

a

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theorem and.elim_right {a b : Prop} (h : a ∧ b) :

b

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@[nolint]

def rfl {α : Sort u} {a : α} :

a = a

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theorem eq.subst {α : Sort u} {P : α → Prop} {a b : α} (h₁ : a = b) (h₂ : P a) :

P b

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@[trans]

theorem eq.trans {α : Sort u} {a b c : α} (h₁ : a = b) (h₂ : b = c) :

a = c

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@[symm]

theorem eq.symm {α : Sort u} {a b : α} (h : a = b) :

b = a

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def heq.rfl {α : Sort u} {a : α} :

a == a

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theorem eq_of_heq {α : Sort u} {a a' : α} (h : a == a') :

a = a'

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inductive sum (α : Type u) (β : Type v) :

Type (max u v)

inl : Π {α : Type u} {β : Type v}, α → α ⊕ β
inr : Π {α : Type u} {β : Type v}, β → α ⊕ β

Instances for sum

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inductive psum (α : Sort u) (β : Sort v) :

Sort (max 1 u v)

inl : Π {α : Sort u} {β : Sort v}, α → psum α β
inr : Π {α : Sort u} {β : Sort v}, β → psum α β

Instances for psum

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inductive or (a b : Prop) :

Prop

inl : ∀ {a b : Prop}, a → a ∨ b
inr : ∀ {a b : Prop}, b → a ∨ b

Logical or.

or P Q, with notation P ∨ Q, is the proposition which is true if and only if P or Q is true.

To prove a goal ⊢ P ∨ Q, if you know which alternative you want to prove, you can use the tactics left (which gives the goal ⊢ P) or right (which gives the goal ⊢ Q).

Given a hypothesis h : P ∨ Q and goal ⊢ R, the tactic cases h will give you two copies of the goal ⊢ R, with the hypothesis h : P in the first, and the hypothesis h : Q in the second.

Instances for or

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theorem or.intro_left {a : Prop} (b : Prop) (ha : a) :

a ∨ b

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theorem or.intro_right (a : Prop) {b : Prop} (hb : b) :

a ∨ b

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structure sigma {α : Type u} (β : α → Type v) :

Type (max u v)

fst : α
snd : β self.fst

Instances for sigma

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structure psigma {α : Sort u} (β : α → Sort v) :

Sort (max 1 u v)

fst : α
snd : β self.fst

Instances for psigma

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inductive bool :

Type

ff : bool
tt : bool

Instances for bool

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structure subtype {α : Sort u} (p : α → Prop) :

Sort (max 1 u)

val : α
property : p self.val

Remark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description.

Instances for subtype

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@[class]

inductive decidable (p : Prop) :

Type

is_false : Π {p : Prop}, ¬p → decidable p
is_true : Π {p : Prop}, p → decidable p

Instances of this typeclass

Instances of other typeclasses for decidable

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@[reducible]

def decidable_pred {α : Sort u} (r : α → Prop) :

Sort (max u 1)

Equations

decidable_pred r = Π (a : α), decidable (r a)

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@[reducible]

def decidable_rel {α : Sort u} (r : α → α → Prop) :

Sort (max u 1)

Equations

decidable_rel r = Π (a b : α), decidable (r a b)

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@[reducible]

def decidable_eq (α : Sort u) :

Sort (max u 1)

Equations

decidable_eq α = decidable_rel eq

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inductive option (α : Type u) :

Type u

none : Π {α : Type u}, option α
some : Π {α : Type u}, α → option α

Instances for option

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inductive list (T : Type u) :

Type u

nil : Π {T : Type u}, list T
cons : Π {T : Type u}, T → list T → list T

Instances for list

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inductive nat :

Type

zero : ℕ
succ : ℕ → ℕ

Instances for nat

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structure unification_constraint :

Type (u+1)

α : Type ?
lhs : self.α
rhs : self.α

Instances for unification_constraint

unification_constraint.inhabited

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structure unification_hint :

Type (max (u_1+1) (u_2+1))

pattern : unification_constraint
constraints : list unification_constraint

Instances for unification_hint

unification_hint.inhabited

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@[ext, class]

structure has_zero (α : Type u) :

Type u

zero : α

Instances of this typeclass

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@[class]

structure has_one (α : Type u) :

Type u

one : α

Instances of this typeclass

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@[class]

structure has_add (α : Type u) :

Type u

add : α → α → α

Instances of this typeclass

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@[class]

structure has_mul (α : Type u) :

Type u

mul : α → α → α

Instances of this typeclass

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@[class]

structure has_inv (α : Type u) :

Type u

inv : α → α

Instances of this typeclass

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@[class]

structure has_neg (α : Type u) :

Type u

neg : α → α

Instances of this typeclass

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@[class]

structure has_sub (α : Type u) :

Type u

sub : α → α → α

Instances of this typeclass

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@[class]

structure has_div (α : Type u) :

Type u

div : α → α → α

Instances of this typeclass

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@[class]

structure has_dvd (α : Type u) :

Type u

dvd : α → α → Prop

Instances of this typeclass

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@[class]

structure has_mod (α : Type u) :

Type u

mod : α → α → α

Instances of this typeclass

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@[ext, class]

structure has_le (α : Type u) :

Type u

le : α → α → Prop

Instances of this typeclass

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@[class]

structure has_lt (α : Type u) :

Type u

lt : α → α → Prop

Instances of this typeclass

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@[class]

structure has_append (α : Type u) :

Type u

append : α → α → α

Instances of this typeclass

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@[class]

structure has_andthen (α : Type u) (β : Type v) (σ : out_param (Type w)) :

Type (max u v w)

andthen : α → β → σ

Instances of this typeclass

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@[class]

structure has_union (α : Type u) :

Type u

union : α → α → α

Instances of this typeclass

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@[class]

structure has_inter (α : Type u) :

Type u

inter : α → α → α

Instances of this typeclass

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@[class]

structure has_sdiff (α : Type u) :

Type u

sdiff : α → α → α

Instances of this typeclass

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@[class]

structure has_equiv (α : Sort u) :

Sort (max 1 u)

equiv : α → α → Prop

Instances of this typeclass

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@[class]

structure has_subset (α : Type u) :

Type u

subset : α → α → Prop

Instances of this typeclass

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@[class]

structure has_ssubset (α : Type u) :

Type u

ssubset : α → α → Prop

Instances of this typeclass

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@[class]

structure has_emptyc (α : Type u) :

Type u

emptyc : α

Instances of this typeclass

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@[class]

structure has_insert (α : out_param (Type u)) (γ : Type v) :

Type (max u v)

insert : α → γ → γ

Instances of this typeclass

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@[class]

structure has_singleton (α : out_param (Type u)) (β : Type v) :

Type (max u v)

singleton : α → β

Instances of this typeclass

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@[class]

structure has_sep (α : out_param (Type u)) (γ : Type v) :

Type (max u v)

sep : (α → Prop) → γ → γ

Type class used to implement the notation { a ∈ c | p a }

Instances of this typeclass

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@[class]

structure has_mem (α : out_param (Type u)) (γ : Type v) :

Type (max u v)

mem : α → γ → Prop

Type class for set-like membership

Instances of this typeclass

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@[class]

structure has_pow (α : Type u) (β : Type v) :

Type (max u v)

pow : α → β → α

Instances of this typeclass

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@[reducible]

def ge {α : Type u} [has_le α] (a b : α) :

Prop

Equations

a ≥ b = (b ≤ a)

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@[reducible]

def gt {α : Type u} [has_lt α] (a b : α) :

Prop

Equations

a > b = (b < a)

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@[reducible]

def superset {α : Type u} [has_subset α] (a b : α) :

Prop

Equations

a ⊇ b = (b ⊆ a)

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@[reducible]

def ssuperset {α : Type u} [has_ssubset α] (a b : α) :

Prop

Equations

a ⊃ b = (b ⊂ a)

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def bit0 {α : Type u} [s : has_add α] (a : α) :

α

Equations

bit0 a = a + a

Instances for bit0

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def bit1 {α : Type u} [s₁ : has_one α] [s₂ : has_add α] (a : α) :

α

Equations

bit1 a = bit0 a + 1

Instances for bit1

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@[class]

structure is_lawful_singleton (α : Type u) (β : Type v) [has_emptyc β] [has_insert α β] [has_singleton α β] :

Prop

insert_emptyc_eq : ∀ (x : α), {x} = {x}

Instances of this typeclass

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@[protected]

def nat.add :

ℕ → ℕ → ℕ

Equations

a.add b.succ = (a.add b).succ
a.add 0 = a

source

@[protected, instance]

def nat.has_zero :

has_zero ℕ

Equations

nat.has_zero = {zero := 0}

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@[protected, instance]

def nat.has_one :

has_one ℕ

Equations

nat.has_one = {one := 1}

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@[protected, instance]

def nat.has_add :

has_add ℕ

Equations

nat.has_add = {add := nat.add}

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def std.priority.default :

ℕ

Equations

std.priority.default = 1000

source

def std.priority.max :

ℕ

Equations

std.priority.max = 4294967295

source

@[protected]

def nat.prio :

ℕ

Equations

nat.prio = std.priority.default + 100

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def std.prec.max :

ℕ

Equations

std.prec.max = 1024

source

def std.prec.arrow :

ℕ

Equations

std.prec.arrow = 25

source

def std.prec.max_plus :

ℕ

This def is "max + 10". It can be used e.g. for postfix operations that should be stronger than application.

Equations

std.prec.max_plus = std.prec.max + 10

source

@[class]

structure has_sizeof (α : Sort u) :

Sort (max 1 u)

sizeof : α → ℕ

Instances of this typeclass

default_has_sizeof
slim_check.sampleable_ext.wf
slim_check.sampleable.wf
nat.has_sizeof
prod.has_sizeof
sum.has_sizeof
psum.has_sizeof
sigma.has_sizeof
psigma.has_sizeof
punit.has_sizeof
bool.has_sizeof
option.has_sizeof
list.has_sizeof
subtype.has_sizeof
bin_tree.has_sizeof_inst
inhabited.has_sizeof_inst
ulift.has_sizeof_inst
plift.has_sizeof_inst
has_well_founded.has_sizeof_inst
setoid.has_sizeof_inst
char.has_sizeof_inst
char.has_sizeof
string_imp.has_sizeof_inst
string.iterator_imp.has_sizeof_inst
string.has_sizeof
fin.has_sizeof_inst
has_repr.has_sizeof_inst
ordering.has_sizeof_inst
has_lift.has_sizeof_inst
has_lift_t.has_sizeof_inst
has_coe.has_sizeof_inst
has_coe_t.has_sizeof_inst
has_coe_to_fun.has_sizeof_inst
has_coe_to_sort.has_sizeof_inst
has_coe_t_aux.has_sizeof_inst
has_to_string.has_sizeof_inst
name.has_sizeof_inst
functor.has_sizeof_inst
has_pure.has_sizeof_inst
has_seq.has_sizeof_inst
has_seq_left.has_sizeof_inst
has_seq_right.has_sizeof_inst
applicative.has_sizeof_inst
has_orelse.has_sizeof_inst
alternative.has_sizeof_inst
has_bind.has_sizeof_inst
monad.has_sizeof_inst
has_monad_lift.has_sizeof_inst
has_monad_lift_t.has_sizeof_inst
monad_functor.has_sizeof_inst
monad_functor_t.has_sizeof_inst
monad_run.has_sizeof_inst
format.color.has_sizeof_inst
monad_fail.has_sizeof_inst
pos.has_sizeof_inst
binder_info.has_sizeof_inst
reducibility_hints.has_sizeof_inst
environment.projection_info.has_sizeof_inst
environment.implicit_infer_kind.has_sizeof_inst
tactic.transparency.has_sizeof_inst
tactic.new_goals.has_sizeof_inst
tactic.apply_cfg.has_sizeof_inst
param_info.has_sizeof_inst
fun_info.has_sizeof_inst
subsingleton_info.has_sizeof_inst
occurrences.has_sizeof_inst
tactic.rewrite_cfg.has_sizeof_inst
tactic.dsimp_config.has_sizeof_inst
tactic.dunfold_config.has_sizeof_inst
tactic.delta_config.has_sizeof_inst
tactic.unfold_proj_config.has_sizeof_inst
tactic.simp_config.has_sizeof_inst
tactic.simp_intros_config.has_sizeof_inst
interactive.loc.has_sizeof_inst
cc_config.has_sizeof_inst
congr_arg_kind.has_sizeof_inst
tactic.interactive.case_tag.has_sizeof_inst
tactic.interactive.case_tag.match_result.has_sizeof_inst
tactic.unfold_config.has_sizeof_inst
except.has_sizeof_inst
except_t.has_sizeof_inst
monad_except.has_sizeof_inst
monad_except_adapter.has_sizeof_inst
state_t.has_sizeof_inst
monad_state.has_sizeof_inst
monad_state_adapter.has_sizeof_inst
reader_t.has_sizeof_inst
monad_reader.has_sizeof_inst
monad_reader_adapter.has_sizeof_inst
option_t.has_sizeof_inst
vm_obj_kind.has_sizeof_inst
vm_decl_kind.has_sizeof_inst
ematch_config.has_sizeof_inst
smt_pre_config.has_sizeof_inst
smt_config.has_sizeof_inst
rsimp.config.has_sizeof_inst
expr.coord.has_sizeof_inst
preorder.has_sizeof_inst
partial_order.has_sizeof_inst
linear_order.has_sizeof_inst
int.has_sizeof_inst
d_array.has_sizeof_inst
widget.mouse_event_kind.has_sizeof_inst
feature_search.feature_cfg.has_sizeof_inst
feature_search.predictor_type.has_sizeof_inst
feature_search.predictor_cfg.has_sizeof_inst
dlist.has_sizeof_inst
doc_category.has_sizeof_inst
tactic_doc_entry.has_sizeof_inst
pempty.has_sizeof_inst
rbnode.has_sizeof_inst
rbnode.color.has_sizeof_inst
random_gen.has_sizeof_inst
std_gen.has_sizeof_inst
io.error.has_sizeof_inst
io.mode.has_sizeof_inst
io.process.stdio.has_sizeof_inst
io.process.spawn_args.has_sizeof_inst
monad_io.has_sizeof_inst
monad_io_terminal.has_sizeof_inst
monad_io_net_system.has_sizeof_inst
monad_io_file_system.has_sizeof_inst
monad_io_environment.has_sizeof_inst
monad_io_process.has_sizeof_inst
monad_io_random.has_sizeof_inst
function.has_uncurry.has_sizeof_inst
to_additive.value_type.has_sizeof_inst
lint_verbosity.has_sizeof_inst
tactic.explode.status.has_sizeof_inst
auto.auto_config.has_sizeof_inst
auto.case_option.has_sizeof_inst
tactic.itauto.and_kind.has_sizeof_inst
tactic.itauto.prop.has_sizeof_inst
tactic.itauto.proof.has_sizeof_inst
can_lift.has_sizeof_inst
norm_cast.label.has_sizeof_inst
_nest_1_1._nest_1_1.tactic.tactic_script._mut_.has_sizeof_inst
_nest_1_1.tactic.tactic_script.has_sizeof_inst
_nest_1_1.list.tactic.tactic_script.has_sizeof_inst
tactic.tactic_script.has_sizeof_inst
simps_cfg.has_sizeof_inst
applicative_transformation.has_sizeof_inst
traversable.has_sizeof_inst
is_lawful_traversable.has_sizeof_inst
tactic.suggest.head_symbol_match.has_sizeof_inst
has_vadd.has_sizeof_inst
has_vsub.has_sizeof_inst
has_smul.has_sizeof_inst
semigroup.has_sizeof_inst
add_semigroup.has_sizeof_inst
comm_semigroup.has_sizeof_inst
add_comm_semigroup.has_sizeof_inst
left_cancel_semigroup.has_sizeof_inst
add_left_cancel_semigroup.has_sizeof_inst
right_cancel_semigroup.has_sizeof_inst
add_right_cancel_semigroup.has_sizeof_inst
mul_one_class.has_sizeof_inst
add_zero_class.has_sizeof_inst
add_monoid.has_sizeof_inst
monoid.has_sizeof_inst
add_comm_monoid.has_sizeof_inst
comm_monoid.has_sizeof_inst
add_left_cancel_monoid.has_sizeof_inst
left_cancel_monoid.has_sizeof_inst
add_right_cancel_monoid.has_sizeof_inst
right_cancel_monoid.has_sizeof_inst
add_cancel_monoid.has_sizeof_inst
cancel_monoid.has_sizeof_inst
add_cancel_comm_monoid.has_sizeof_inst
cancel_comm_monoid.has_sizeof_inst
has_involutive_neg.has_sizeof_inst
has_involutive_inv.has_sizeof_inst
div_inv_monoid.has_sizeof_inst
sub_neg_monoid.has_sizeof_inst
neg_zero_class.has_sizeof_inst
sub_neg_zero_monoid.has_sizeof_inst
inv_one_class.has_sizeof_inst
div_inv_one_monoid.has_sizeof_inst
subtraction_monoid.has_sizeof_inst
division_monoid.has_sizeof_inst
subtraction_comm_monoid.has_sizeof_inst
division_comm_monoid.has_sizeof_inst
group.has_sizeof_inst
add_group.has_sizeof_inst
comm_group.has_sizeof_inst
add_comm_group.has_sizeof_inst
unique.has_sizeof_inst
mul_zero_class.has_sizeof_inst
semigroup_with_zero.has_sizeof_inst
mul_zero_one_class.has_sizeof_inst
monoid_with_zero.has_sizeof_inst
cancel_monoid_with_zero.has_sizeof_inst
comm_monoid_with_zero.has_sizeof_inst
cancel_comm_monoid_with_zero.has_sizeof_inst
group_with_zero.has_sizeof_inst
comm_group_with_zero.has_sizeof_inst
fun_like.has_sizeof_inst
embedding_like.has_sizeof_inst
equiv_like.has_sizeof_inst
equiv.has_sizeof_inst
has_compl.has_sizeof_inst
has_sup.has_sizeof_inst
has_inf.has_sizeof_inst
units.has_sizeof_inst
add_units.has_sizeof_inst
zero_hom.has_sizeof_inst
zero_hom_class.has_sizeof_inst
add_hom.has_sizeof_inst
add_hom_class.has_sizeof_inst
add_monoid_hom.has_sizeof_inst
add_monoid_hom_class.has_sizeof_inst
one_hom.has_sizeof_inst
one_hom_class.has_sizeof_inst
mul_hom.has_sizeof_inst
mul_hom_class.has_sizeof_inst
monoid_hom.has_sizeof_inst
monoid_hom_class.has_sizeof_inst
monoid_with_zero_hom.has_sizeof_inst
monoid_with_zero_hom_class.has_sizeof_inst
function.embedding.has_sizeof_inst
add_action.has_sizeof_inst
mul_action.has_sizeof_inst
smul_zero_class.has_sizeof_inst
distrib_smul.has_sizeof_inst
distrib_mul_action.has_sizeof_inst
mul_distrib_mul_action.has_sizeof_inst
rat.has_sizeof_inst
has_nat_cast.has_sizeof_inst
add_monoid_with_one.has_sizeof_inst
add_comm_monoid_with_one.has_sizeof_inst
has_int_cast.has_sizeof_inst
add_group_with_one.has_sizeof_inst
add_comm_group_with_one.has_sizeof_inst
distrib.has_sizeof_inst
left_distrib_class.has_sizeof_inst
right_distrib_class.has_sizeof_inst
non_unital_non_assoc_semiring.has_sizeof_inst
non_unital_semiring.has_sizeof_inst
non_assoc_semiring.has_sizeof_inst
semiring.has_sizeof_inst
non_unital_comm_semiring.has_sizeof_inst
comm_semiring.has_sizeof_inst
has_distrib_neg.has_sizeof_inst
non_unital_non_assoc_ring.has_sizeof_inst
non_unital_ring.has_sizeof_inst
non_assoc_ring.has_sizeof_inst
ring.has_sizeof_inst
non_unital_comm_ring.has_sizeof_inst
comm_ring.has_sizeof_inst
has_rat_cast.has_sizeof_inst
division_semiring.has_sizeof_inst
division_ring.has_sizeof_inst
semifield.has_sizeof_inst
field.has_sizeof_inst
equiv_functor.has_sizeof_inst
is_nonstrict_strict_order.has_sizeof_inst
rel_hom.has_sizeof_inst
rel_hom_class.has_sizeof_inst
rel_embedding.has_sizeof_inst
rel_iso.has_sizeof_inst
semilattice_sup.has_sizeof_inst
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