statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
combine {p q : Prop} : psum unit (p → q) → psum unit p → psum unit q | | (psum.inr f) (psum.inr x) := psum.inr (f x)
| _ _ := psum.inl () | def | slim_check.combine | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | applicative combinator proof carrying test results | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
and_counter_example {p q : Prop} :
test_result p →
test_result q →
test_result (p ∧ q) | | (failure Hce xs n) _ := failure (λ h, Hce h.1) xs n
| _ (failure Hce xs n) := failure (λ h, Hce h.2) xs n
| (success xs) (success ys) := success $ combine (combine (psum.inr and.intro) xs) ys
| (gave_up n) (gave_up m) := gave_up $ n + m
| (gave_up n) _ := gave_up n
| _ (gave_up n) := gave_up n | def | slim_check.and_counter_example | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Combine the test result for properties `p` and `q` to create a test for their conjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
or_counter_example {p q : Prop} :
test_result p →
test_result q →
test_result (p ∨ q) | | (failure Hce xs n) (failure Hce' ys n') := failure (λ h, or_iff_not_and_not.1 h ⟨Hce, Hce'⟩)
(xs ++ ys) (n + n')
| (success xs) _ := success $ combine (psum.inr or.inl) xs
| _ (success ys) := success $ combine (psum.inr or.inr) ys
| (gave_up n) (gave_up m) := gave_up $ n + m
| (gave_up n) _ := gave_up n
| _ (gave_u... | def | slim_check.or_counter_example | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Combine the test result for properties `p` and `q` to create a test for their disjunction | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convert_counter_example {p q : Prop}
(h : q → p) :
test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q | | (failure Hce xs n) _ := failure (mt h Hce) xs n
| (success Hp) Hpq := success (combine Hpq Hp)
| (gave_up n) _ := gave_up n | def | slim_check.convert_counter_example | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | If `q → p`, then `¬ p → ¬ q` which means that testing `p` can allow us
to find counter-examples to `q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convert_counter_example' {p q : Prop}
(h : p ↔ q) (r : test_result p) :
test_result q | convert_counter_example h.2 r (psum.inr h.1) | def | slim_check.convert_counter_example' | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Test `q` by testing `p` and proving the equivalence between the two. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_to_counter_example (x : string) {p q : Prop}
(h : q → p) :
test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q | | (failure Hce xs n) _ := failure (mt h Hce) (x :: xs) n
| r hpq := convert_counter_example h r hpq | def | slim_check.add_to_counter_example | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | When we assign a value to a universally quantified variable,
we record that value using this function so that our counter-examples
can be informative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_var_to_counter_example {γ : Type v} [has_repr γ]
(var : string) (x : γ) {p q : Prop}
(h : q → p) : test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q | @add_to_counter_example (var ++ " := " ++ repr x) _ _ h | def | slim_check.add_var_to_counter_example | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Add some formatting to the information recorded by `add_to_counter_example`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
named_binder (n : string) (p : Prop) : Prop | p | def | slim_check.named_binder | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Gadget used to introspect the name of bound variables.
It is used with the `testable` typeclass so that
`testable (named_binder "x" (∀ x, p x))` can use the variable name
of `x` in error messages displayed to the user. If we find that instantiating
the above quantifier with 3 falsifies it, we can print:
```
=========... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_failure {p} : test_result p → bool | | (test_result.failure _ _ _) := tt
| _ := ff | def | slim_check.is_failure | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Is the given test result a failure? | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
and_testable (p q : Prop) [testable p] [testable q] :
testable (p ∧ q) | ⟨ λ cfg min, do
xp ← testable.run p cfg min,
xq ← testable.run q cfg min,
pure $ and_counter_example xp xq ⟩ | instance | slim_check.and_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
or_testable (p q : Prop) [testable p] [testable q] :
testable (p ∨ q) | ⟨ λ cfg min, do
xp ← testable.run p cfg min,
match xp with
| success (psum.inl h) := pure $ success (psum.inl h)
| success (psum.inr h) := pure $ success (psum.inr $ or.inl h)
| _ := do
xq ← testable.run q cfg min,
pure $ or_counter_example xp xq
end ⟩ | instance | slim_check.or_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_testable (p q : Prop) [testable ((p ∧ q) ∨ (¬ p ∧ ¬ q))] :
testable (p ↔ q) | ⟨ λ cfg min, do
xp ← testable.run ((p ∧ q) ∨ (¬ p ∧ ¬ q)) cfg min,
return $ convert_counter_example' (by tauto!) xp ⟩ | instance | slim_check.iff_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dec_guard_testable (p : Prop) [printable_prop p] [decidable p] (β : p → Prop)
[∀ h, testable (β h)] : testable (named_binder var $ Π h, β h) | ⟨ λ cfg min, do
if h : p
then
match print_prop p with
| none := (λ r, convert_counter_example ($ h) r (psum.inr $ λ q _, q)) <$>
testable.run (β h) cfg min
| some str := (λ r, add_to_counter_example (sformat!"guard: {str}") ($ h) r
(psum.inr $ λ q _, q)) <$> testable.run (β h) ... | instance | slim_check.dec_guard_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
use_has_to_string (α : Type*) | α | def | slim_check.use_has_to_string | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Type tag that replaces a type's `has_repr` instance with its `has_to_string` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
use_has_to_string.inhabited [I : inhabited α] : inhabited (use_has_to_string α) | I | instance | slim_check.use_has_to_string.inhabited | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
use_has_to_string.mk {α} (x : α) : use_has_to_string α | x | def | slim_check.use_has_to_string.mk | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Add the type tag `use_has_to_string` to an expression's type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
all_types_testable [testable (f ℤ)] : testable (named_binder var $ Π x, f x) | ⟨ λ cfg min, do
r ← testable.run (f ℤ) cfg min,
return $ add_var_to_counter_example var (use_has_to_string.mk "ℤ") ($ ℤ) r ⟩ | instance | slim_check.all_types_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_if_giveup {p α β} [has_repr α] (tracing_enabled : bool) (var : string) (val : α) :
test_result p → thunk β → β | | (test_result.gave_up _) :=
if tracing_enabled then trace (sformat!" {var} := {repr val}")
else ($ ())
| _ := ($ ()) | def | slim_check.trace_if_giveup | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Trace the value of sampled variables if the sample is discarded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
test_forall_in_list
[∀ x, testable (β x)] [has_repr α] :
Π xs : list α, testable (named_binder var $ ∀ x, named_binder var' $ x ∈ xs → β x) | | [] := ⟨ λ tracing min, return $ success $ psum.inr (by { introv x h, cases h} ) ⟩
| (x :: xs) :=
⟨ λ cfg min, do
r ← testable.run (β x) cfg min,
trace_if_giveup cfg.trace_discarded var x r $
match r with
| failure _ _ _ := return $ add_var_to_counter_example var x
... | instance | slim_check.test_forall_in_list | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"not_and_of_not_right"
] | testable instance for a property iterating over the element of a list | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combine_testable (p : Prop)
(t : list $ testable p) (h : 0 < t.length) : testable p | ⟨ λ cfg min, have 0 < length (map (λ t, @testable.run _ t cfg min) t),
by { rw [length_map], apply h },
gen.one_of (list.map (λ t, @testable.run _ t cfg min) t) this ⟩ | def | slim_check.combine_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Test proposition `p` by randomly selecting one of the provided
testable instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
format_failure (s : string) (xs : list string) (n : ℕ) : string | let counter_ex := string.intercalate "\n" xs in
sformat!"
===================
{s}
{counter_ex}
({n} shrinks)
-------------------
" | def | slim_check.format_failure | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Format the counter-examples found in a test failure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
format_failure' (s : string) {p} : test_result p → string | | (success a) := ""
| (gave_up a) := ""
| (test_result.failure _ xs n) := format_failure s xs n | def | slim_check.format_failure' | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Format the counter-examples found in a test failure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_shrinks {p} (n : ℕ) : test_result p → test_result p | | r@(success a) := r
| r@(gave_up a) := r
| (test_result.failure h vs n') := test_result.failure h vs $ n + n' | def | slim_check.add_shrinks | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Increase the number of shrinking steps in a test result. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minimize_aux [sampleable_ext α] [∀ x, testable (β x)] (cfg : slim_check_cfg) (var : string) :
proxy_repr α → ℕ → option_t gen (Σ x, test_result (β (interp α x))) | well_founded.fix has_well_founded.wf $ λ x f_rec n, do
if cfg.trace_shrink_candidates
then return $ trace sformat!
"candidates for {var} :=\n{repr (sampleable_ext.shrink x).to_list}\n" ()
else pure (),
⟨y,r,⟨h₁⟩⟩ ← (sampleable_ext.shrink x).mfirst (λ ⟨a,h⟩, do
⟨r⟩ ← monad_lift (uliftable.up $ test... | def | slim_check.minimize_aux | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"uliftable.up"
] | Shrink a counter-example `x` by using `shrink x`, picking the first
candidate that falsifies a property and recursively shrinking that one.
The process is guaranteed to terminate because `shrink x` produces
a proof that all the values it produces are smaller (according to `sizeof`)
than `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minimize [sampleable_ext α] [∀ x, testable (β x)] (cfg : slim_check_cfg) (var : string)
(x : proxy_repr α) (r : test_result (β (interp α x))) :
gen (Σ x, test_result (β (interp α x))) | do
if cfg.trace_shrink then return $
trace (sformat!"{var} := {repr x}" ++ format_failure' "Shrink counter-example:" r) ()
else pure (),
x' ← option_t.run $ minimize_aux α _ cfg var x 0,
pure $ x'.get_or_else ⟨x, r⟩ | def | slim_check.minimize | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Once a property fails to hold on an example, look for smaller counter-examples
to show the user. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_testable (p : Prop)
[testable (named_binder var (∀ x, named_binder var' $ β x → p))] :
testable (named_binder var' (named_binder var (∃ x, β x) → p)) | ⟨ λ cfg min, do
x ← testable.run (named_binder var (∀ x, named_binder var' $ β x → p)) cfg min,
pure $ convert_counter_example' exists_imp_distrib.symm x ⟩ | instance | slim_check.exists_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
var_testable [sampleable_ext α] [∀ x, testable (β x)] :
testable (named_binder var $ Π x : α, β x) | ⟨ λ cfg min, do
uliftable.adapt_down (sampleable_ext.sample α) $
λ x, do
r ← testable.run (β (sampleable_ext.interp α x)) cfg ff,
uliftable.adapt_down (if is_failure r ∧ min
then minimize _ _ cfg var x r
else if cfg.trace_success
... | instance | slim_check.var_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"uliftable.adapt_down"
] | Test a universal property by creating a sample of the right type and instantiating the
bound variable with it | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prop_var_testable (β : Prop → Prop) [I : ∀ b : bool, testable (β b)] :
testable (named_binder var $ Π p : Prop, β p) | ⟨λ cfg min, do
convert_counter_example (λ h (b : bool), h b) <$> @testable.run
(named_binder var $ Π b : bool, β b) _ cfg min⟩ | instance | slim_check.prop_var_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Test a universal property about propositions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unused_var_testable (β) [inhabited α] [testable β] :
testable (named_binder var $ Π x : α, β) | ⟨ λ cfg min, do
r ← testable.run β cfg min,
pure $ convert_counter_example ($ default) r (psum.inr $ λ x _, x) ⟩ | instance | slim_check.unused_var_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_var_testable {p : α → Prop}
[∀ x, printable_prop (p x)]
[∀ x, testable (β x)]
[I : sampleable_ext (subtype p)] :
testable (named_binder var $ Π x : α, named_binder var' $ p x → β x) | ⟨ λ cfg min,
do let test (x : subtype p) : testable (β x) :=
⟨ λ cfg min, do
r ← testable.run (β x.val) cfg min,
match print_prop (p x) with
| none := pure r
| some str := pure $ add_to_counter_example sformat!"guard: {str} (by construction)"
... | instance | slim_check.subtype_var_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"slim_check.var_testable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_testable (p : Prop) [printable_prop p] [decidable p] : testable p | ⟨ λ cfg min, return $
if h : p then success (psum.inr h)
else
match print_prop p with
| none := failure h [] 0
| some str := failure h [sformat!"issue: {str} does not hold"] 0
end ⟩ | instance | slim_check.decidable_testable | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq.printable_prop {α} [has_repr α] (x y : α) : printable_prop (x = y) | ⟨ some sformat!"{repr x} = {repr y}" ⟩ | instance | slim_check.eq.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne.printable_prop {α} [has_repr α] (x y : α) : printable_prop (x ≠ y) | ⟨ some sformat!"{repr x} ≠ {repr y}" ⟩ | instance | slim_check.ne.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le.printable_prop {α} [has_le α] [has_repr α] (x y : α) : printable_prop (x ≤ y) | ⟨ some sformat!"{repr x} ≤ {repr y}" ⟩ | instance | slim_check.le.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt.printable_prop {α} [has_lt α] [has_repr α] (x y : α) : printable_prop (x < y) | ⟨ some sformat!"{repr x} < {repr y}" ⟩ | instance | slim_check.lt.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
perm.printable_prop {α} [has_repr α] (xs ys : list α) : printable_prop (xs ~ ys) | ⟨ some sformat!"{repr xs} ~ {repr ys}" ⟩ | instance | slim_check.perm.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
and.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ∧ y) | ⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ∧ {y'})" ⟩ | instance | slim_check.and.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
or.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ∨ y) | ⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ∨ {y'})" ⟩ | instance | slim_check.or.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ↔ y) | ⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ↔ {y'})" ⟩ | instance | slim_check.iff.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
imp.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x → y) | ⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} → {y'})" ⟩ | instance | slim_check.imp.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not.printable_prop (x : Prop) [printable_prop x] : printable_prop (¬ x) | ⟨ do x' ← print_prop x,
some sformat!"¬ {x'}" ⟩ | instance | slim_check.not.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
true.printable_prop : printable_prop true | ⟨ some "true" ⟩ | instance | slim_check.true.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
false.printable_prop : printable_prop false | ⟨ some "false" ⟩ | instance | slim_check.false.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bool.printable_prop (b : bool) : printable_prop b | ⟨ some $ if b then "true" else "false" ⟩ | instance | slim_check.bool.printable_prop | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
retry (cmd : rand (test_result p)) : ℕ → rand (test_result p) | | 0 := return $ gave_up 1
| (succ n) := do
r ← cmd,
match r with
| success hp := return $ success hp
| (failure Hce xs n) := return (failure Hce xs n)
| (gave_up _) := retry n
end | def | slim_check.retry | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"rand"
] | Execute `cmd` and repeat every time the result is `gave_up` (at most
`n` times). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
give_up (x : ℕ) : test_result p → test_result p | | (success (psum.inl ())) := gave_up x
| (success (psum.inr p)) := success (psum.inr p)
| (gave_up n) := gave_up (n+x)
| (failure Hce xs n) := failure Hce xs n | def | slim_check.give_up | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Count the number of times the test procedure gave up. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
testable.run_suite_aux (cfg : slim_check_cfg) : test_result p → ℕ → rand (test_result p) | | r 0 := return r
| r (succ n) :=
do let size := (cfg.num_inst - n - 1) * cfg.max_size / cfg.num_inst,
when cfg.trace_success $ return $ trace sformat!"[slim_check: sample]" (),
x ← retry ( (testable.run p cfg tt).run ⟨ size ⟩) 10,
match x with
| (success (psum.inl ())) := testable.run_suite_aux r n
| (s... | def | slim_check.testable.run_suite_aux | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"rand"
] | Try `n` times to find a counter-example for `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
testable.run_suite (cfg : slim_check_cfg := {}) : rand (test_result p) | testable.run_suite_aux p cfg (success $ psum.inl ()) cfg.num_inst | def | slim_check.testable.run_suite | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"rand"
] | Try to find a counter-example of `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
testable.check' (cfg : slim_check_cfg := {}) : io (test_result p) | match cfg.random_seed with
| some seed := io.run_rand_with seed (testable.run_suite p cfg)
| none := io.run_rand (testable.run_suite p cfg)
end | def | slim_check.testable.check' | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"io.run_rand",
"io.run_rand_with"
] | Run a test suite for `p` in `io`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_existential_decorations : expr → expr | | e@`(@Exists %%α %%(lam n bi d b)) :=
let n := to_string n in
const ``named_binder [] (`(n) : expr) e
| e := e | def | slim_check.tactic.add_existential_decorations | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | `add_existential_decorations p` adds `a `named_binder` annotation at the
root of `p` if `p` is an existential quantification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_decorations : expr → expr | e | e.replace $ λ e _,
match e with
| (pi n bi d b) :=
let n := to_string n in
some $ const ``named_binder [] (`(n) : expr)
(pi n bi (add_existential_decorations d) (add_decorations b))
| e := none
end | def | slim_check.tactic.add_decorations | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | Traverse the syntax of a proposition to find universal quantifiers
and existential quantifiers and add `named_binder` annotations next to
them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decorations_of (p : Prop) | Prop | def | slim_check.tactic.decorations_of | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | `decorations_of p` is used as a hint to `mk_decorations` to specify
that the goal should be satisfied with a proposition equivalent to `p`
with added annotations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_decorations : tactic unit | do
`(tactic.decorations_of %%p) ← target,
exact $ add_decorations p | def | slim_check.tactic.mk_decorations | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [] | In a goal of the shape `⊢ tactic.decorations_of p`, `mk_decoration` examines
the syntax of `p` and add `named_binder` around universal quantifications and
existential quantifications to improve error messages.
This tool can be used in the declaration of a function as follows:
```lean
def foo (p : Prop) (p' : tactic.d... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
testable.check (p : Prop) (cfg : slim_check_cfg := {})
(p' : tactic.decorations_of p . tactic.mk_decorations) [testable p'] : io punit | do
x ← match cfg.random_seed with
| some seed := io.run_rand_with seed (testable.run_suite p' cfg)
| none := io.run_rand (testable.run_suite p' cfg)
end,
match x with
| (success _) := when (¬ cfg.quiet) $ io.put_str_ln "Success"
| (gave_up n) := io.fail sformat!"Gave up {repr n} times"
| (failure _ xs n) :=... | def | slim_check.testable.check | testing.slim_check | src/testing/slim_check/testable.lean | [
"testing.slim_check.sampleable"
] | [
"io.run_rand",
"io.run_rand_with"
] | Run a test suite for `p` and return true or false: should we believe that `p` holds? | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alexandroff (X : Type*) | option X | def | alexandroff | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [] | The Alexandroff extension of an arbitrary topological space `X` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infty : alexandroff X | none | def | alexandroff.infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | The point at infinity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite [infinite X] : infinite (alexandroff X) | option.infinite | instance | alexandroff.infinite | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : X → alexandroff X) | option.some_injective X | lemma | alexandroff.coe_injective | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"option.some_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_coe {x y : X} : (x : alexandroff X) = y ↔ x = y | coe_injective.eq_iff | lemma | alexandroff.coe_eq_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_infty (x : X) : (x : alexandroff X) ≠ ∞ | lemma | alexandroff.coe_ne_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
infty_ne_coe (x : X) : ∞ ≠ (x : alexandroff X) | lemma | alexandroff.infty_ne_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
rec (C : alexandroff X → Sort*) (h₁ : C ∞) (h₂ : Π x : X, C x) :
Π (z : alexandroff X), C z | option.rec h₁ h₂ | def | alexandroff.rec | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | Recursor for `alexandroff` using the preferred forms `∞` and `↑x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compl_range_coe_infty : is_compl (range (coe : X → alexandroff X)) {∞} | is_compl_range_some_none X | lemma | alexandroff.is_compl_range_coe_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coe_union_infty : (range (coe : X → alexandroff X) ∪ {∞}) = univ | range_some_union_none X | lemma | alexandroff.range_coe_union_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coe_inter_infty : (range (coe : X → alexandroff X) ∩ {∞}) = ∅ | range_some_inter_none X | lemma | alexandroff.range_coe_inter_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_range_coe : (range (coe : X → alexandroff X))ᶜ = {∞} | compl_range_some X | lemma | alexandroff.compl_range_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_infty : ({∞}ᶜ : set (alexandroff X)) = range (coe : X → alexandroff X) | (@is_compl_range_coe_infty X).symm.compl_eq | lemma | alexandroff.compl_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_image_coe (s : set X) : (coe '' s : set (alexandroff X))ᶜ = coe '' sᶜ ∪ {∞} | by rw [coe_injective.compl_image_eq, compl_range_coe] | lemma | alexandroff.compl_image_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_infty_iff_exists {x : alexandroff X} :
x ≠ ∞ ↔ ∃ (y : X), (y : alexandroff X) = x | by induction x using alexandroff.rec; simp | lemma | alexandroff.ne_infty_iff_exists | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"alexandroff.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift : can_lift (alexandroff X) X coe (λ x, x ≠ ∞) | with_top.can_lift | instance | alexandroff.can_lift | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"can_lift",
"with_top.can_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_range_coe_iff {x : alexandroff X} :
x ∉ range (coe : X → alexandroff X) ↔ x = ∞ | by rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] | lemma | alexandroff.not_mem_range_coe_iff | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_not_mem_range_coe : ∞ ∉ range (coe : X → alexandroff X) | not_mem_range_coe_iff.2 rfl | lemma | alexandroff.infty_not_mem_range_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_not_mem_image_coe {s : set X} : ∞ ∉ (coe : X → alexandroff X) '' s | not_mem_subset (image_subset_range _ _) infty_not_mem_range_coe | lemma | alexandroff.infty_not_mem_image_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_preimage_infty : (coe : X → alexandroff X) ⁻¹' {∞} = ∅ | by { ext, simp } | lemma | alexandroff.coe_preimage_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_def :
is_open s ↔ (∞ ∈ s → is_compact (coe ⁻¹' s : set X)ᶜ) ∧ is_open (coe ⁻¹' s : set X) | iff.rfl | lemma | alexandroff.is_open_def | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_of_mem' (h : ∞ ∈ s) :
is_open s ↔ is_compact (coe ⁻¹' s : set X)ᶜ ∧ is_open (coe ⁻¹' s : set X) | by simp [is_open_def, h] | lemma | alexandroff.is_open_iff_of_mem' | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_of_mem (h : ∞ ∈ s) :
is_open s ↔ is_closed (coe ⁻¹' s : set X)ᶜ ∧ is_compact (coe ⁻¹' s : set X)ᶜ | by simp only [is_open_iff_of_mem' h, is_closed_compl_iff, and.comm] | lemma | alexandroff.is_open_iff_of_mem | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_closed",
"is_closed_compl_iff",
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_of_not_mem (h : ∞ ∉ s) :
is_open s ↔ is_open (coe ⁻¹' s : set X) | by simp [is_open_def, h] | lemma | alexandroff.is_open_iff_of_not_mem | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_of_mem (h : ∞ ∈ s) :
is_closed s ↔ is_closed (coe ⁻¹' s : set X) | have ∞ ∉ sᶜ, from λ H, H h,
by rw [← is_open_compl_iff, is_open_iff_of_not_mem this, ← is_open_compl_iff, preimage_compl] | lemma | alexandroff.is_closed_iff_of_mem | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_closed",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_of_not_mem (h : ∞ ∉ s) :
is_closed s ↔ is_closed (coe ⁻¹' s : set X) ∧ is_compact (coe ⁻¹' s : set X) | by rw [← is_open_compl_iff, is_open_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl] | lemma | alexandroff.is_closed_iff_of_not_mem | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"compl_compl",
"is_closed",
"is_compact",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_image_coe {s : set X} :
is_open (coe '' s : set (alexandroff X)) ↔ is_open s | by rw [is_open_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective] | lemma | alexandroff.is_open_image_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_compl_image_coe {s : set X} :
is_open (coe '' s : set (alexandroff X))ᶜ ↔ is_closed s ∧ is_compact s | begin
rw [is_open_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective],
exact infty_not_mem_image_coe
end | lemma | alexandroff.is_open_compl_image_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"compl_compl",
"is_closed",
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_image_coe {s : set X} :
is_closed (coe '' s : set (alexandroff X)) ↔ is_closed s ∧ is_compact s | by rw [← is_open_compl_iff, is_open_compl_image_coe] | lemma | alexandroff.is_closed_image_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_closed",
"is_compact",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_of_compl (s : set X) (h₁ : is_closed s) (h₂ : is_compact s) :
topological_space.opens (alexandroff X) | ⟨(coe '' s)ᶜ, is_open_compl_image_coe.2 ⟨h₁, h₂⟩⟩ | def | alexandroff.opens_of_compl | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_closed",
"is_compact",
"topological_space.opens"
] | An open set in `alexandroff X` constructed from a closed compact set in `X` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infty_mem_opens_of_compl {s : set X} (h₁ : is_closed s) (h₂ : is_compact s) :
∞ ∈ opens_of_compl s h₁ h₂ | mem_compl infty_not_mem_image_coe | lemma | alexandroff.infty_mem_opens_of_compl | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : X → alexandroff X) | continuous_def.mpr (λ s hs, hs.right) | lemma | alexandroff.continuous_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_coe : is_open_map (coe : X → alexandroff X) | λ s, is_open_image_coe.2 | lemma | alexandroff.is_open_map_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_coe : open_embedding (coe : X → alexandroff X) | open_embedding_of_continuous_injective_open continuous_coe coe_injective is_open_map_coe | lemma | alexandroff.open_embedding_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_range_coe : is_open (range (coe : X → alexandroff X)) | open_embedding_coe.open_range | lemma | alexandroff.is_open_range_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_infty : is_closed ({∞} : set (alexandroff X)) | by { rw [← compl_range_coe, is_closed_compl_iff], exact is_open_range_coe } | lemma | alexandroff.is_closed_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_closed",
"is_closed_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_coe_eq (x : X) : 𝓝 ↑x = map (coe : X → alexandroff X) (𝓝 x) | (open_embedding_coe.map_nhds_eq x).symm | lemma | alexandroff.nhds_coe_eq | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_coe_image (s : set X) (x : X) :
𝓝[coe '' s] (x : alexandroff X) = map coe (𝓝[s] x) | (open_embedding_coe.to_embedding.map_nhds_within_eq _ _).symm | lemma | alexandroff.nhds_within_coe_image | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_coe (s : set (alexandroff X)) (x : X) :
𝓝[s] ↑x = map coe (𝓝[coe ⁻¹' s] x) | (open_embedding_coe.map_nhds_within_preimage_eq _ _).symm | lemma | alexandroff.nhds_within_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_nhds (x : X) : comap (coe : X → alexandroff X) (𝓝 x) = 𝓝 x | (open_embedding_coe.to_inducing.nhds_eq_comap x).symm | lemma | alexandroff.comap_coe_nhds | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_compl_coe_ne_bot (x : X) [h : ne_bot (𝓝[≠] x)] :
ne_bot (𝓝[≠] (x : alexandroff X)) | by simpa [nhds_within_coe, preimage, coe_eq_coe] using h.map coe | instance | alexandroff.nhds_within_compl_coe_ne_bot | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | If `x` is not an isolated point of `X`, then `x : alexandroff X` is not an isolated point
of `alexandroff X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within_compl_infty_eq : 𝓝[≠] (∞ : alexandroff X) = map coe (coclosed_compact X) | begin
refine (nhds_within_basis_open ∞ _).ext (has_basis_coclosed_compact.map _) _ _,
{ rintro s ⟨hs, hso⟩,
refine ⟨_, (is_open_iff_of_mem hs).mp hso, _⟩,
simp },
{ rintro s ⟨h₁, h₂⟩,
refine ⟨_, ⟨mem_compl infty_not_mem_image_coe, is_open_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩,
simp [compl_image_coe, ← d... | lemma | alexandroff.nhds_within_compl_infty_eq | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"nhds_within_basis_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_compl_infty_ne_bot [noncompact_space X] :
ne_bot (𝓝[≠] (∞ : alexandroff X)) | by { rw nhds_within_compl_infty_eq, apply_instance } | instance | alexandroff.nhds_within_compl_infty_ne_bot | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"noncompact_space"
] | If `X` is a non-compact space, then `∞` is not an isolated point of `alexandroff X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within_compl_ne_bot [∀ x : X, ne_bot (𝓝[≠] x)] [noncompact_space X]
(x : alexandroff X) : ne_bot (𝓝[≠] x) | alexandroff.rec _ alexandroff.nhds_within_compl_infty_ne_bot
(λ y, alexandroff.nhds_within_compl_coe_ne_bot y) x | instance | alexandroff.nhds_within_compl_ne_bot | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"alexandroff.nhds_within_compl_coe_ne_bot",
"alexandroff.nhds_within_compl_infty_ne_bot",
"alexandroff.rec",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_infty_eq : 𝓝 (∞ : alexandroff X) = map coe (coclosed_compact X) ⊔ pure ∞ | by rw [← nhds_within_compl_infty_eq, nhds_within_compl_singleton_sup_pure] | lemma | alexandroff.nhds_infty_eq | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"nhds_within_compl_singleton_sup_pure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_infty :
(𝓝 (∞ : alexandroff X)).has_basis (λ s : set X, is_closed s ∧ is_compact s)
(λ s, coe '' sᶜ ∪ {∞}) | begin
rw nhds_infty_eq,
exact (has_basis_coclosed_compact.map _).sup_pure _
end | lemma | alexandroff.has_basis_nhds_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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