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 |
|---|---|---|---|---|---|---|---|---|---|---|
to_measure_of_fintype_apply [measurable_space α] (hs : measurable_set s) :
(of_fintype f h).to_measure s = ∑' x, s.indicator f x | (to_measure_apply_eq_to_outer_measure_apply _ s hs).trans
(to_outer_measure_of_fintype_apply h s) | lemma | pmf.to_measure_of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize (f : α → ℝ≥0∞) (hf0 : tsum f ≠ 0) (hf : tsum f ≠ ∞) : pmf α | ⟨λ a, f a * (∑' x, f x)⁻¹, ennreal.summable.has_sum_iff.2
(ennreal.tsum_mul_right.trans (ennreal.mul_inv_cancel hf0 hf))⟩ | def | pmf.normalize | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"ennreal.mul_inv_cancel",
"normalize",
"pmf",
"tsum"
] | Given a `f` with non-zero and non-infinite sum, get a `pmf` by normalizing `f` by its `tsum` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_apply (a : α) : (normalize f hf0 hf) a = f a * (∑' x, f x)⁻¹ | rfl | lemma | pmf.normalize_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"normalize",
"normalize_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_normalize : (normalize f hf0 hf).support = function.support f | set.ext (λ a, by simp [hf, mem_support_iff]) | lemma | pmf.support_normalize | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"function.support",
"normalize",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_normalize_iff (a : α) : a ∈ (normalize f hf0 hf).support ↔ f a ≠ 0 | by simp | lemma | pmf.mem_support_normalize_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter (p : pmf α) (s : set α) (h : ∃ a ∈ s, a ∈ p.support) : pmf α | pmf.normalize (s.indicator p) (by simpa using h) (p.tsum_coe_indicator_ne_top s) | def | pmf.filter | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"filter",
"pmf",
"pmf.normalize"
] | Create new `pmf` by filtering on a set with non-zero measure and normalizing | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter_apply (a : α) : (p.filter s h) a = (s.indicator p a) * (∑' a', (s.indicator p) a')⁻¹ | by rw [filter, normalize_apply] | lemma | pmf.filter_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"filter",
"normalize_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_apply_eq_zero_of_not_mem {a : α} (ha : a ∉ s) : (p.filter s h) a = 0 | by rw [filter_apply, set.indicator_apply_eq_zero.mpr (λ ha', absurd ha' ha), zero_mul] | lemma | pmf.filter_apply_eq_zero_of_not_mem | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_filter_iff {a : α} : a ∈ (p.filter s h).support ↔ a ∈ s ∧ a ∈ p.support | (mem_support_normalize_iff _ _ _).trans set.indicator_apply_ne_zero | lemma | pmf.mem_support_filter_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_filter : (p.filter s h).support = s ∩ p.support | set.ext $ λ x, (mem_support_filter_iff _) | lemma | pmf.support_filter | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_apply_eq_zero_iff (a : α) : (p.filter s h) a = 0 ↔ a ∉ s ∨ a ∉ p.support | by erw [apply_eq_zero_iff, support_filter, set.mem_inter_iff, not_and_distrib] | lemma | pmf.filter_apply_eq_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"not_and_distrib",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_apply_ne_zero_iff (a : α) : (p.filter s h) a ≠ 0 ↔ a ∈ s ∧ a ∈ p.support | by rw [ne.def, filter_apply_eq_zero_iff, not_or_distrib, not_not, not_not] | lemma | pmf.filter_apply_ne_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"not_not",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bernoulli (p : ℝ≥0∞) (h : p ≤ 1) : pmf bool | of_fintype (λ b, cond b p (1 - p)) (by simp [h]) | def | pmf.bernoulli | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"bernoulli",
"pmf"
] | A `pmf` which assigns probability `p` to `tt` and `1 - p` to `ff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bernoulli_apply : bernoulli p h b = cond b p (1 - p) | rfl | lemma | pmf.bernoulli_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"bernoulli"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_bernoulli : (bernoulli p h).support = {b | cond b (p ≠ 0) (p ≠ 1)} | begin
refine set.ext (λ b, _),
induction b,
{ simp_rw [mem_support_iff, bernoulli_apply, bool.cond_ff, ne.def, tsub_eq_zero_iff_le, not_le],
exact ⟨ne_of_lt, lt_of_le_of_ne h⟩ },
{ simp only [mem_support_iff, bernoulli_apply, bool.cond_tt, set.mem_set_of_eq], }
end | lemma | pmf.support_bernoulli | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"bernoulli",
"bool.cond_ff",
"bool.cond_tt",
"set.ext",
"tsub_eq_zero_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_bernoulli_iff : b ∈ (bernoulli p h).support ↔ cond b (p ≠ 0) (p ≠ 1) | by simp | lemma | pmf.mem_support_bernoulli_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"bernoulli"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure (a : α) : pmf α | ⟨λ a', if a' = a then 1 else 0, has_sum_ite_eq _ _⟩ | def | pmf.pure | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"has_sum_ite_eq",
"pmf"
] | The pure `pmf` is the `pmf` where all the mass lies in one point.
The value of `pure a` is `1` at `a` and `0` elsewhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pure_apply : pure a a' = (if a' = a then 1 else 0) | rfl | lemma | pmf.pure_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_pure : (pure a).support = {a} | set.ext (λ a', by simp [mem_support_iff]) | lemma | pmf.support_pure | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_pure_iff: a' ∈ (pure a).support ↔ a' = a | by simp | lemma | pmf.mem_support_pure_iff | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_apply_self : pure a a = 1 | if_pos rfl | lemma | pmf.pure_apply_self | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_apply_of_ne (h : a' ≠ a) : pure a a' = 0 | if_neg h | lemma | pmf.pure_apply_of_ne | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_pure_apply : (pure a).to_outer_measure s = if a ∈ s then 1 else 0 | begin
refine (to_outer_measure_apply (pure a) s).trans _,
split_ifs with ha ha,
{ refine ((tsum_congr (λ b, _)).trans (tsum_ite_eq a 1)),
exact ite_eq_left_iff.2 (λ hb, symm (ite_eq_right_iff.2 (λ h, (hb $ h.symm ▸ ha).elim))) },
{ refine ((tsum_congr (λ b, _)).trans (tsum_zero)),
exact ite_eq_right_iff... | lemma | pmf.to_outer_measure_pure_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"tsum_congr",
"tsum_ite_eq",
"tsum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_pure_apply (hs : measurable_set s) :
(pure a).to_measure s = if a ∈ s then 1 else 0 | (to_measure_apply_eq_to_outer_measure_apply (pure a) s hs).trans (to_outer_measure_pure_apply a s) | lemma | pmf.to_measure_pure_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"measurable_set"
] | The measure of a set under `pure a` is `1` for sets containing `a` and `0` otherwise | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_measure_pure : (pure a).to_measure = measure.dirac a | measure.ext (λ s hs, by simpa only [to_measure_pure_apply a s hs, measure.dirac_apply' a hs]) | lemma | pmf.to_measure_pure | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pmf_dirac [countable α] [h : measurable_singleton_class α] :
(measure.dirac a).to_pmf = pure a | by rw [to_pmf_eq_iff_to_measure_eq, to_measure_pure] | lemma | pmf.to_pmf_dirac | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"countable",
"measurable_singleton_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind (p : pmf α) (f : α → pmf β) : pmf β | ⟨λ b, ∑' a, p a * f a b, ennreal.summable.has_sum_iff.2 (ennreal.tsum_comm.trans $
by simp only [ennreal.tsum_mul_left, tsum_coe, mul_one])⟩ | def | pmf.bind | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_mul_left",
"mul_one",
"pmf"
] | The monadic bind operation for `pmf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind_apply (b : β) : p.bind f b = ∑'a, p a * f a b | rfl | lemma | pmf.bind_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_bind : (p.bind f).support = ⋃ a ∈ p.support, (f a).support | set.ext (λ b, by simp [mem_support_iff, ennreal.tsum_eq_zero, not_or_distrib]) | lemma | pmf.support_bind | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_eq_zero",
"not_or_distrib",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_bind_iff (b : β) : b ∈ (p.bind f).support ↔ ∃ a ∈ p.support, b ∈ (f a).support | by simp only [support_bind, set.mem_Union, set.mem_set_of_eq] | lemma | pmf.mem_support_bind_iff | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_bind (a : α) (f : α → pmf β) : (pure a).bind f = f a | have ∀ b a', ite (a' = a) 1 0 * f a' b = ite (a' = a) (f a b) 0, from
assume b a', by split_ifs; simp; subst h; simp,
by ext b; simp [this] | lemma | pmf.pure_bind | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_pure : p.bind pure = p | pmf.ext (λ x, (bind_apply _ _ _).trans (trans (tsum_eq_single x $
(λ y hy, by rw [pure_apply_of_ne _ _ hy.symm, mul_zero])) $ by rw [pure_apply_self, mul_one])) | lemma | pmf.bind_pure | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"mul_one",
"mul_zero",
"pmf.ext",
"tsum_eq_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_const (p : pmf α) (q : pmf β) : p.bind (λ _, q) = q | pmf.ext (λ x, by rw [bind_apply, ennreal.tsum_mul_right, tsum_coe, one_mul]) | lemma | pmf.bind_const | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_mul_right",
"one_mul",
"pmf",
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_bind : (p.bind f).bind g = p.bind (λ a, (f a).bind g) | pmf.ext (λ b, by simpa only [ennreal.coe_eq_coe.symm, bind_apply, ennreal.tsum_mul_left.symm,
ennreal.tsum_mul_right.symm, mul_assoc, mul_left_comm, mul_comm] using ennreal.tsum_comm) | lemma | pmf.bind_bind | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_comm",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_comm (p : pmf α) (q : pmf β) (f : α → β → pmf γ) :
p.bind (λ a, q.bind (f a)) = q.bind (λ b, p.bind (λ a, f a b)) | pmf.ext (λ b, by simpa only [ennreal.coe_eq_coe.symm, bind_apply, ennreal.tsum_mul_left.symm,
ennreal.tsum_mul_right.symm, mul_assoc, mul_left_comm, mul_comm] using ennreal.tsum_comm) | lemma | pmf.bind_comm | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_comm",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"pmf",
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_bind_apply :
(p.bind f).to_outer_measure s = ∑' a, p a * (f a).to_outer_measure s | calc (p.bind f).to_outer_measure s
= ∑' b, if b ∈ s then ∑' a, p a * f a b else 0 :
by simp [to_outer_measure_apply, set.indicator_apply]
... = ∑' b a, p a * (if b ∈ s then f a b else 0) :
tsum_congr (λ b, by split_ifs; simp)
... = ∑' a b, p a * (if b ∈ s then f a b else 0) :
tsum_comm' ennreal.summab... | lemma | pmf.to_outer_measure_bind_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.summable",
"ennreal.tsum_mul_left",
"tsum_comm'",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_bind_apply [measurable_space β] (hs : measurable_set s) :
(p.bind f).to_measure s = ∑' a, p a * (f a).to_measure s | (to_measure_apply_eq_to_outer_measure_apply (p.bind f) s hs).trans
((to_outer_measure_bind_apply p f s).trans (tsum_congr (λ a, congr_arg (λ x, p a * x)
(to_measure_apply_eq_to_outer_measure_apply (f a) s hs).symm))) | lemma | pmf.to_measure_bind_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"measurable_set",
"measurable_space",
"tsum_congr"
] | The measure of a set under `p.bind f` is the sum over `a : α`
of the probability of `a` under `p` times the measure of the set under `f a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind_on_support (p : pmf α) (f : Π a ∈ p.support, pmf β) : pmf β | ⟨λ b, ∑' a, p a * if h : p a = 0 then 0 else f a h b,
ennreal.summable.has_sum_iff.2 begin
refine (ennreal.tsum_comm.trans (trans (tsum_congr $ λ a, _) p.tsum_coe)),
simp_rw [ennreal.tsum_mul_left],
split_ifs with h,
{ simp only [h, zero_mul] },
{ rw [(f a h).tsum_coe, mul_one] }
end⟩ | def | pmf.bind_on_support | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_mul_left",
"mul_one",
"pmf",
"tsum_congr",
"zero_mul"
] | Generalized version of `bind` allowing `f` to only be defined on the support of `p`.
`p.bind f` is equivalent to `p.bind_on_support (λ a _, f a)`, see `bind_on_support_eq_bind` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind_on_support_apply (b : β) :
p.bind_on_support f b = ∑' a, p a * if h : p a = 0 then 0 else f a h b | rfl | lemma | pmf.bind_on_support_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_bind_on_support :
(p.bind_on_support f).support = ⋃ (a : α) (h : a ∈ p.support), (f a h).support | begin
refine set.ext (λ b, _),
simp only [ennreal.tsum_eq_zero, not_or_distrib, mem_support_iff,
bind_on_support_apply, ne.def, not_forall, mul_eq_zero, set.mem_Union],
exact ⟨λ hb, let ⟨a, ⟨ha, ha'⟩⟩ := hb in ⟨a, ha, by simpa [ha] using ha'⟩,
λ hb, let ⟨a, ha, ha'⟩ := hb in ⟨a, ⟨ha, by simpa [(mem_suppor... | lemma | pmf.support_bind_on_support | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_eq_zero",
"mul_eq_zero",
"not_forall",
"not_or_distrib",
"set.ext",
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_bind_on_support_iff (b : β) :
b ∈ (p.bind_on_support f).support ↔ ∃ (a : α) (h : a ∈ p.support), b ∈ (f a h).support | by simp only [support_bind_on_support, set.mem_set_of_eq, set.mem_Union] | lemma | pmf.mem_support_bind_on_support_iff | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_on_support_eq_bind (p : pmf α) (f : α → pmf β) :
p.bind_on_support (λ a _, f a) = p.bind f | begin
ext b x,
have : ∀ a, ite (p a = 0) 0 (p a * f a b) = p a * f a b,
from λ a, ite_eq_right_iff.2 (λ h, h.symm ▸ symm (zero_mul $ f a b)),
simp only [bind_on_support_apply (λ a _, f a), p.bind_apply f,
dite_eq_ite, mul_ite, mul_zero, this],
end | lemma | pmf.bind_on_support_eq_bind | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"dite_eq_ite",
"mul_ite",
"mul_zero",
"pmf",
"zero_mul"
] | `bind_on_support` reduces to `bind` if `f` doesn't depend on the additional hypothesis | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind_on_support_eq_zero_iff (b : β) :
p.bind_on_support f b = 0 ↔ ∀ a (ha : p a ≠ 0), f a ha b = 0 | begin
simp only [bind_on_support_apply, ennreal.tsum_eq_zero, mul_eq_zero, or_iff_not_imp_left],
exact ⟨λ h a ha, trans (dif_neg ha).symm (h a ha), λ h a ha, trans (dif_neg ha) (h a ha)⟩,
end | lemma | pmf.bind_on_support_eq_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_eq_zero",
"mul_eq_zero",
"or_iff_not_imp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_bind_on_support (a : α) (f : Π (a' : α) (ha : a' ∈ (pure a).support), pmf β) :
(pure a).bind_on_support f = f a ((mem_support_pure_iff a a).mpr rfl) | begin
refine pmf.ext (λ b, _),
simp only [bind_on_support_apply, pure_apply],
refine trans (tsum_congr (λ a', _)) (tsum_ite_eq a _),
by_cases h : (a' = a); simp [h],
end | lemma | pmf.pure_bind_on_support | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"pmf",
"pmf.ext",
"tsum_congr",
"tsum_ite_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_on_support_pure (p : pmf α) :
p.bind_on_support (λ a _, pure a) = p | by simp only [pmf.bind_pure, pmf.bind_on_support_eq_bind] | lemma | pmf.bind_on_support_pure | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"pmf",
"pmf.bind_on_support_eq_bind",
"pmf.bind_pure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_on_support_bind_on_support (p : pmf α)
(f : ∀ a ∈ p.support, pmf β)
(g : ∀ (b ∈ (p.bind_on_support f).support), pmf γ) :
(p.bind_on_support f).bind_on_support g =
p.bind_on_support (λ a ha, (f a ha).bind_on_support
(λ b hb, g b ((mem_support_bind_on_support_iff f b).mpr ⟨a, ha, hb⟩))) | begin
refine pmf.ext (λ a, _),
simp only [ennreal.coe_eq_coe.symm, bind_on_support_apply, ← tsum_dite_right,
ennreal.tsum_mul_left.symm, ennreal.tsum_mul_right.symm],
simp only [ennreal.tsum_eq_zero, ennreal.coe_eq_coe, ennreal.coe_eq_zero, ennreal.coe_zero,
dite_eq_left_iff, mul_eq_zero],
refine ennrea... | lemma | pmf.bind_on_support_bind_on_support | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"dite_eq_left_iff",
"ennreal.coe_eq_coe",
"ennreal.coe_eq_zero",
"ennreal.coe_zero",
"ennreal.tsum_eq_zero",
"mul_eq_zero",
"pmf",
"pmf.ext",
"tsum_congr",
"tsum_dite_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_on_support_comm (p : pmf α) (q : pmf β)
(f : ∀ (a ∈ p.support) (b ∈ q.support), pmf γ) :
p.bind_on_support (λ a ha, q.bind_on_support (f a ha)) =
q.bind_on_support (λ b hb, p.bind_on_support (λ a ha, f a ha b hb)) | begin
apply pmf.ext, rintro c,
simp only [ennreal.coe_eq_coe.symm, bind_on_support_apply, ← tsum_dite_right,
ennreal.tsum_mul_left.symm, ennreal.tsum_mul_right.symm],
refine trans (ennreal.tsum_comm) (tsum_congr (λ b, tsum_congr (λ a, _))),
split_ifs with h1 h2 h2; ring,
end | lemma | pmf.bind_on_support_comm | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_comm",
"pmf",
"pmf.ext",
"ring",
"tsum_congr",
"tsum_dite_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_bind_on_support_apply : (p.bind_on_support f).to_outer_measure s
= ∑' a, p a * if h : p a = 0 then 0 else (f a h).to_outer_measure s | begin
simp only [to_outer_measure_apply, set.indicator_apply, bind_on_support_apply],
calc ∑' b, ite (b ∈ s) (∑' a, p a * dite (p a = 0) (λ h, 0) (λ h, f a h b)) 0
= ∑' b a, ite (b ∈ s) (p a * dite (p a = 0) (λ h, 0) (λ h, f a h b)) 0 :
tsum_congr (λ b, by split_ifs with hbs; simp only [eq_self_iff_true, ... | lemma | pmf.to_outer_measure_bind_on_support_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"ennreal.tsum_comm",
"ennreal.tsum_mul_left",
"mul_ite",
"mul_zero",
"tsum_congr",
"tsum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_bind_on_support_apply [measurable_space β] (hs : measurable_set s) :
(p.bind_on_support f).to_measure s
= ∑' a, p a * if h : p a = 0 then 0 else (f a h).to_measure s | by simp only [to_measure_apply_eq_to_outer_measure_apply _ _ hs,
to_outer_measure_bind_on_support_apply] | lemma | pmf.to_measure_bind_on_support_apply | probability.probability_mass_function | src/probability/probability_mass_function/monad.lean | [
"probability.probability_mass_function.basic"
] | [
"measurable_set",
"measurable_space"
] | The measure of a set under `p.bind_on_support f` is the sum over `a : α`
of the probability of `a` under `p` times the measure of the set under `f a _`.
The additional if statement is needed since `f` is only a partial function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_of_finset (s : finset α) (hs : s.nonempty) : pmf α | of_finset (λ a, if a ∈ s then s.card⁻¹ else 0) s (Exists.rec_on hs (λ x hx,
calc ∑ (a : α) in s, ite (a ∈ s) (s.card : ℝ≥0∞)⁻¹ 0
= ∑ (a : α) in s, (s.card : ℝ≥0∞)⁻¹ : finset.sum_congr rfl (λ x hx, by simp [hx])
... = (s.card : ℝ≥0∞) * (s.card : ℝ≥0∞)⁻¹ : by rw [finset.sum_const, nsmul_eq_mul]
... = 1 : en... | def | pmf.uniform_of_finset | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"ennreal.mul_inv_cancel",
"ennreal.nat_ne_top",
"finset",
"finset.card_eq_zero",
"nat.cast_eq_zero",
"nsmul_eq_mul",
"pmf"
] | Uniform distribution taking the same non-zero probability on the nonempty finset `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_of_finset_apply (a : α) :
uniform_of_finset s hs a = if a ∈ s then s.card⁻¹ else 0 | rfl | lemma | pmf.uniform_of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_of_finset_apply_of_mem (ha : a ∈ s) : uniform_of_finset s hs a = (s.card)⁻¹ | by simp [ha] | lemma | pmf.uniform_of_finset_apply_of_mem | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_of_finset_apply_of_not_mem (ha : a ∉ s) : uniform_of_finset s hs a = 0 | by simp [ha] | lemma | pmf.uniform_of_finset_apply_of_not_mem | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_uniform_of_finset : (uniform_of_finset s hs).support = s | set.ext (let ⟨a, ha⟩ := hs in by simp [mem_support_iff, finset.ne_empty_of_mem ha]) | lemma | pmf.support_uniform_of_finset | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"finset.ne_empty_of_mem",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_uniform_of_finset_iff (a : α) : a ∈ (uniform_of_finset s hs).support ↔ a ∈ s | by simp | lemma | pmf.mem_support_uniform_of_finset_iff | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_uniform_of_finset_apply :
(uniform_of_finset s hs).to_outer_measure t = (s.filter (∈ t)).card / s.card | calc (uniform_of_finset s hs).to_outer_measure t
= ∑' x, if x ∈ t then (uniform_of_finset s hs x) else 0 :
to_outer_measure_apply (uniform_of_finset s hs) t
... = ∑' x, if x ∈ s ∧ x ∈ t then (s.card : ℝ≥0∞)⁻¹ else 0 :
(tsum_congr (λ x, by simp only [uniform_of_finset_apply,
and_comm (x ∈ s), ite_and, ... | lemma | pmf.to_outer_measure_uniform_of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"div_eq_mul_inv",
"ennreal.coe_nat",
"finset.card_ne_zero_of_mem",
"ite_and",
"nsmul_eq_mul",
"tsum_congr",
"tsum_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_uniform_of_finset_apply [measurable_space α] (ht : measurable_set t) :
(uniform_of_finset s hs).to_measure t = (s.filter (∈ t)).card / s.card | (to_measure_apply_eq_to_outer_measure_apply _ t ht).trans
(to_outer_measure_uniform_of_finset_apply hs t) | lemma | pmf.to_measure_uniform_of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_of_fintype (α : Type*) [fintype α] [nonempty α] : pmf α | uniform_of_finset (finset.univ) (finset.univ_nonempty) | def | pmf.uniform_of_fintype | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"finset.univ",
"finset.univ_nonempty",
"fintype",
"pmf"
] | The uniform pmf taking the same uniform value on all of the fintype `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_of_fintype_apply (a : α) : uniform_of_fintype α a = (fintype.card α)⁻¹ | by simpa only [uniform_of_fintype, finset.mem_univ, if_true, uniform_of_finset_apply] | lemma | pmf.uniform_of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"finset.mem_univ",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_uniform_of_fintype (α : Type*) [fintype α] [nonempty α] :
(uniform_of_fintype α).support = ⊤ | set.ext (λ x, by simp [mem_support_iff]) | lemma | pmf.support_uniform_of_fintype | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"fintype",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_uniform_of_fintype (a : α) : a ∈ (uniform_of_fintype α).support | by simp | lemma | pmf.mem_support_uniform_of_fintype | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_uniform_of_fintype_apply :
(uniform_of_fintype α).to_outer_measure s = fintype.card s / fintype.card α | by simpa [uniform_of_fintype] | lemma | pmf.to_outer_measure_uniform_of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_uniform_of_fintype_apply [measurable_space α] (hs : measurable_set s) :
(uniform_of_fintype α).to_measure s = fintype.card s / fintype.card α | by simpa [uniform_of_fintype, hs] | lemma | pmf.to_measure_uniform_of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"fintype.card",
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_multiset (s : multiset α) (hs : s ≠ 0) : pmf α | ⟨λ a, s.count a / s.card, ennreal.summable.has_sum_iff.2
(calc ∑' (b : α), (s.count b : ℝ≥0∞) / s.card = s.card⁻¹ * ∑' b, s.count b :
by simp_rw [ennreal.div_eq_inv_mul, ennreal.tsum_mul_left]
... = s.card⁻¹ * ∑ b in s.to_finset, (s.count b : ℝ≥0∞) :
congr_arg (λ x, s.card⁻¹ * x) (tsum_eq_sum $ λ a ha... | def | pmf.of_multiset | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"ennreal.div_eq_inv_mul",
"ennreal.inv_mul_cancel",
"ennreal.nat_ne_top",
"ennreal.tsum_mul_left",
"multiset",
"multiset.count_eq_zero",
"multiset.mem_to_finset",
"multiset.to_finset_sum_count_eq",
"nat.cast_sum",
"pmf",
"tsum_eq_sum"
] | Given a non-empty multiset `s` we construct the `pmf` which sends `a` to the fraction of
elements in `s` that are `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_multiset_apply (a : α) : of_multiset s hs a = s.count a / s.card | rfl | lemma | pmf.of_multiset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_of_multiset : (of_multiset s hs).support = s.to_finset | set.ext (by simp [mem_support_iff, hs]) | lemma | pmf.support_of_multiset | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_of_multiset_iff (a : α) : a ∈ (of_multiset s hs).support ↔ a ∈ s.to_finset | by simp | lemma | pmf.mem_support_of_multiset_iff | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_multiset_apply_of_not_mem {a : α} (ha : a ∉ s) : of_multiset s hs a = 0 | by simpa only [of_multiset_apply, ennreal.div_zero_iff, nat.cast_eq_zero,
multiset.count_eq_zero, ennreal.nat_ne_top, or_false] using ha | lemma | pmf.of_multiset_apply_of_not_mem | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"ennreal.div_zero_iff",
"ennreal.nat_ne_top",
"multiset.count_eq_zero",
"nat.cast_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_of_multiset_apply :
(of_multiset s hs).to_outer_measure t = (∑' x, (s.filter (∈ t)).count x) / s.card | begin
rw [div_eq_mul_inv, ← ennreal.tsum_mul_right, to_outer_measure_apply],
refine tsum_congr (λ x, _),
by_cases hx : x ∈ t;
simp [set.indicator, hx, div_eq_mul_inv],
end | lemma | pmf.to_outer_measure_of_multiset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"div_eq_mul_inv",
"ennreal.tsum_mul_right",
"set.indicator",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_of_multiset_apply [measurable_space α] (ht : measurable_set t) :
(of_multiset s hs).to_measure t = (∑' x, (s.filter (∈ t)).count x) / s.card | (to_measure_apply_eq_to_outer_measure_apply _ t ht).trans
(to_outer_measure_of_multiset_apply hs t) | lemma | pmf.to_measure_of_multiset_apply | probability.probability_mass_function | src/probability/probability_mass_function/uniform.lean | [
"probability.probability_mass_function.constructions"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted (f : filtration ι m) (u : ι → Ω → β) : Prop | ∀ i : ι, strongly_measurable[f i] (u i) | def | measure_theory.adapted | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | A sequence of functions `u` is adapted to a filtration `f` if for all `i`,
`u i` is `f i`-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul [has_mul β] [has_continuous_mul β]
(hu : adapted f u) (hv : adapted f v) :
adapted f (u * v) | λ i, (hu i).mul (hv i) | lemma | measure_theory.adapted.mul | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div [has_div β] [has_continuous_div β]
(hu : adapted f u) (hv : adapted f v) :
adapted f (u / v) | λ i, (hu i).div (hv i) | lemma | measure_theory.adapted.div | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"has_continuous_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv [group β] [topological_group β] (hu : adapted f u) :
adapted f u⁻¹ | λ i, (hu i).inv | lemma | measure_theory.adapted.inv | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul [has_smul ℝ β] [has_continuous_smul ℝ β] (c : ℝ) (hu : adapted f u) :
adapted f (c • u) | λ i, (hu i).const_smul c | lemma | measure_theory.adapted.smul | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"has_continuous_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable {i : ι} (hf : adapted f u) :
strongly_measurable[m] (u i) | (hf i).mono (f.le i) | lemma | measure_theory.adapted.strongly_measurable | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_le {i j : ι} (hf : adapted f u) (hij : i ≤ j) :
strongly_measurable[f j] (u i) | (hf i).mono (f.mono hij) | lemma | measure_theory.adapted.strongly_measurable_le | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_const (f : filtration ι m) (x : β) : adapted f (λ _ _, x) | λ i, strongly_measurable_const | lemma | measure_theory.adapted_const | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_zero [has_zero β] (f : filtration ι m) : adapted f (0 : ι → Ω → β) | λ i, @strongly_measurable_zero Ω β (f i) _ _ | lemma | measure_theory.adapted_zero | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filtration.adapted_natural [metrizable_space β] [mβ : measurable_space β] [borel_space β]
{u : ι → Ω → β} (hum : ∀ i, strongly_measurable[m] (u i)) :
adapted (filtration.natural u hum) u | begin
assume i,
refine strongly_measurable.mono _ (le_supr₂_of_le i (le_refl i) le_rfl),
rw strongly_measurable_iff_measurable_separable,
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).is_separable_range⟩
end | lemma | measure_theory.filtration.adapted_natural | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"borel_space",
"le_rfl",
"le_supr₂_of_le",
"measurable_space",
"strongly_measurable_iff_measurable_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable [measurable_space ι] (f : filtration ι m) (u : ι → Ω → β) : Prop | ∀ i, strongly_measurable[subtype.measurable_space.prod (f i)] (λ p : set.Iic i × Ω, u p.1 p.2) | def | measure_theory.prog_measurable | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"measurable_space",
"set.Iic"
] | Progressively measurable process. A sequence of functions `u` is said to be progressively
measurable with respect to a filtration `f` if at each point in time `i`, `u` restricted to
`set.Iic i × Ω` is measurable with respect to the product `measurable_space` structure where the
σ-algebra used for `Ω` is `f i`.
The usua... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prog_measurable_const [measurable_space ι] (f : filtration ι m) (b : β) :
prog_measurable f ((λ _ _, b) : ι → Ω → β) | λ i, @strongly_measurable_const _ _ (subtype.measurable_space.prod (f i)) _ _ | lemma | measure_theory.prog_measurable_const | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted (h : prog_measurable f u) : adapted f u | begin
intro i,
have : u i = (λ p : set.Iic i × Ω, u p.1 p.2) ∘ (λ x, (⟨i, set.mem_Iic.mpr le_rfl⟩, x)) := rfl,
rw this,
exact (h i).comp_measurable measurable_prod_mk_left,
end | lemma | measure_theory.prog_measurable.adapted | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"measurable_prod_mk_left",
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {t : ι → Ω → ι} [topological_space ι] [borel_space ι] [metrizable_space ι]
(h : prog_measurable f u) (ht : prog_measurable f t)
(ht_le : ∀ i ω, t i ω ≤ i) :
prog_measurable f (λ i ω, u (t i ω) ω) | begin
intro i,
have : (λ p : ↥(set.Iic i) × Ω, u (t (p.fst : ι) p.snd) p.snd)
= (λ p : ↥(set.Iic i) × Ω, u (p.fst : ι) p.snd) ∘ (λ p : ↥(set.Iic i) × Ω,
(⟨t (p.fst : ι) p.snd, set.mem_Iic.mpr ((ht_le _ _).trans p.fst.prop)⟩, p.snd)) := rfl,
rw this,
exact (h i).comp_measurable ((ht i).measurable.subty... | lemma | measure_theory.prog_measurable.comp | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"borel_space",
"measurable_snd",
"set.Iic",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul [has_mul β] [has_continuous_mul β]
(hu : prog_measurable f u) (hv : prog_measurable f v) :
prog_measurable f (λ i ω, u i ω * v i ω) | λ i, (hu i).mul (hv i) | lemma | measure_theory.prog_measurable.mul | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_prod' {γ} [comm_monoid β] [has_continuous_mul β]
{U : γ → ι → Ω → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) :
prog_measurable f (∏ c in s, U c) | finset.prod_induction U (prog_measurable f) (λ _ _, prog_measurable.mul)
(prog_measurable_const _ 1) h | lemma | measure_theory.prog_measurable.finset_prod' | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"comm_monoid",
"finset",
"finset.prod_induction",
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_prod {γ} [comm_monoid β] [has_continuous_mul β]
{U : γ → ι → Ω → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) :
prog_measurable f (λ i a, ∏ c in s, U c i a) | by { convert prog_measurable.finset_prod' h, ext i a, simp only [finset.prod_apply], } | lemma | measure_theory.prog_measurable.finset_prod | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"comm_monoid",
"finset",
"finset.prod_apply",
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv [group β] [topological_group β] (hu : prog_measurable f u) :
prog_measurable f (λ i ω, (u i ω)⁻¹) | λ i, (hu i).inv | lemma | measure_theory.prog_measurable.inv | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div [group β] [topological_group β]
(hu : prog_measurable f u) (hv : prog_measurable f v) :
prog_measurable f (λ i ω, u i ω / v i ω) | λ i, (hu i).div (hv i) | lemma | measure_theory.prog_measurable.div | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable_of_tendsto' {γ} [measurable_space ι] [pseudo_metrizable_space β]
(fltr : filter γ) [fltr.ne_bot] [fltr.is_countably_generated] {U : γ → ι → Ω → β}
(h : ∀ l, prog_measurable f (U l)) (h_tendsto : tendsto U fltr (𝓝 u)) :
prog_measurable f u | begin
assume i,
apply @strongly_measurable_of_tendsto (set.Iic i × Ω) β γ (measurable_space.prod _ (f i))
_ _ fltr _ _ _ _ (λ l, h l i),
rw tendsto_pi_nhds at h_tendsto ⊢,
intro x,
specialize h_tendsto x.fst,
rw tendsto_nhds at h_tendsto ⊢,
exact λ s hs h_mem, h_tendsto {g | g x.snd ∈ s} (hs.preimage (... | lemma | measure_theory.prog_measurable_of_tendsto' | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"continuous_apply",
"filter",
"measurable_space",
"measurable_space.prod",
"set.Iic",
"strongly_measurable_of_tendsto",
"tendsto_nhds",
"tendsto_pi_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable_of_tendsto [measurable_space ι] [pseudo_metrizable_space β]
{U : ℕ → ι → Ω → β}
(h : ∀ l, prog_measurable f (U l)) (h_tendsto : tendsto U at_top (𝓝 u)) :
prog_measurable f u | prog_measurable_of_tendsto' at_top h h_tendsto | lemma | measure_theory.prog_measurable_of_tendsto | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.prog_measurable_of_continuous
[topological_space ι] [metrizable_space ι] [second_countable_topology ι]
[measurable_space ι] [opens_measurable_space ι]
[pseudo_metrizable_space β]
(h : adapted f u) (hu_cont : ∀ ω, continuous (λ i, u i ω)) :
prog_measurable f u | λ i, @strongly_measurable_uncurry_of_continuous_of_strongly_measurable _ _ (set.Iic i) _ _ _ _ _ _ _
(f i) _ (λ ω, (hu_cont ω).comp continuous_induced_dom) (λ j, (h j).mono (f.mono j.prop)) | theorem | measure_theory.adapted.prog_measurable_of_continuous | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"continuous",
"continuous_induced_dom",
"measurable_space",
"opens_measurable_space",
"set.Iic",
"topological_space"
] | A continuous and adapted process is progressively measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adapted.prog_measurable_of_discrete [topological_space ι] [discrete_topology ι]
[second_countable_topology ι] [measurable_space ι] [opens_measurable_space ι]
[pseudo_metrizable_space β]
(h : adapted f u) :
prog_measurable f u | h.prog_measurable_of_continuous (λ _, continuous_of_discrete_topology) | lemma | measure_theory.adapted.prog_measurable_of_discrete | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [
"continuous_of_discrete_topology",
"discrete_topology",
"measurable_space",
"opens_measurable_space",
"topological_space"
] | For filtrations indexed by a discrete order, `adapted` and `prog_measurable` are equivalent.
This lemma provides `adapted f u → prog_measurable f u`.
See `prog_measurable.adapted` for the reverse direction, which is true more generally. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
predictable.adapted {f : filtration ℕ m} {u : ℕ → Ω → β}
(hu : adapted f (λ n, u (n + 1))) (hu0 : strongly_measurable[f 0] (u 0)) :
adapted f u | λ n, match n with
| 0 := hu0
| n + 1 := (hu n).mono (f.mono n.le_succ)
end | lemma | measure_theory.predictable.adapted | probability.process | src/probability/process/adapted.lean | [
"probability.process.filtration",
"topology.instances.discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filtration {Ω : Type*} (ι : Type*) [preorder ι] (m : measurable_space Ω) | (seq : ι → measurable_space Ω)
(mono' : monotone seq)
(le' : ∀ i : ι, seq i ≤ m) | structure | measure_theory.filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_space",
"monotone"
] | A `filtration` on a measurable space `Ω` with σ-algebra `m` is a monotone
sequence of sub-σ-algebras of `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono {i j : ι} (f : filtration ι m) (hij : i ≤ j) : f i ≤ f j | f.mono' hij | lemma | measure_theory.filtration.mono | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le (f : filtration ι m) (i : ι) : f i ≤ m | f.le' i | lemma | measure_theory.filtration.le | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : filtration ι m} (h : (f : ι → measurable_space Ω) = g) : f = g | by { cases f, cases g, simp only, exact h, } | lemma | measure_theory.filtration.ext | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (m' : measurable_space Ω) (hm' : m' ≤ m) : filtration ι m | ⟨λ _, m', monotone_const, λ _, hm'⟩ | def | measure_theory.filtration.const | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_space",
"monotone_const"
] | The constant filtration which is equal to `m` for all `i : ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply {m' : measurable_space Ω} {hm' : m' ≤ m} (i : ι) : const ι m' hm' i = m' | rfl | lemma | measure_theory.filtration.const_apply | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.