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
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measurable_set_stopping_time_le [topological_space ι]
[second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
measurable_set[hπ.measurable_space] {ω | τ ω ≤ π ω} | begin
suffices : measurable_set[(hτ.min hπ).measurable_space] {ω : Ω | τ ω ≤ π ω},
by { rw measurable_set_min_iff hτ hπ at this, exact this.2, },
rw [← set.univ_inter {ω : Ω | τ ω ≤ π ω}, ← hτ.measurable_set_inter_le_iff hπ, set.univ_inter],
exact measurable_set_le_stopping_time hτ hπ,
end | lemma | measure_theory.is_stopping_time.measurable_set_stopping_time_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"measurable_set",
"measurable_space",
"order_topology",
"set.univ_inter",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_stopping_time [add_group ι]
[topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι]
[measurable_singleton_class ι] [second_countable_topology ι] [has_measurable_sub₂ ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
measurable_set[hτ.measurable_space] {ω | τ ω ... | begin
rw hτ.measurable_set,
intro j,
have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j}
= {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j},
{ ext1 ω,
simp only [set.mem_inter_iff, set.mem_set_of_eq],
refine ⟨λ h, ⟨⟨_, h.2⟩, _⟩, λ h, ⟨_, h.1.2⟩⟩,
{ rw h.1, },
{ rw ← h.1, exact h.2, },
... | lemma | measure_theory.is_stopping_time.measurable_set_eq_stopping_time | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_group",
"borel_space",
"has_measurable_sub₂",
"measurable_set",
"measurable_set.inter",
"measurable_set_eq_fun",
"measurable_singleton_class",
"measurable_space",
"order_topology",
"set.mem_inter_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_stopping_time_of_countable [countable ι]
[topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι]
[measurable_singleton_class ι] [second_countable_topology ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
measurable_set[hτ.measurable_space] {ω | τ ω = π ω} | begin
rw hτ.measurable_set,
intro j,
have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j}
= {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j},
{ ext1 ω,
simp only [set.mem_inter_iff, set.mem_set_of_eq],
refine ⟨λ h, ⟨⟨_, h.2⟩, _⟩, λ h, ⟨_, h.1.2⟩⟩,
{ rw h.1, },
{ rw ← h.1, exact h.2, },
... | lemma | measure_theory.is_stopping_time.measurable_set_eq_stopping_time_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"countable",
"measurable_set",
"measurable_set.inter",
"measurable_set_eq_fun_of_countable",
"measurable_singleton_class",
"measurable_space",
"order_topology",
"set.mem_inter_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value (u : ι → Ω → β) (τ : Ω → ι) : Ω → β | λ ω, u (τ ω) ω | def | measure_theory.stopped_value | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | Given a map `u : ι → Ω → E`, its stopped value with respect to the stopping
time `τ` is the map `x ↦ u (τ ω) ω`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_value_const (u : ι → Ω → β) (i : ι) : stopped_value u (λ ω, i) = u i | rfl | lemma | measure_theory.stopped_value_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process (u : ι → Ω → β) (τ : Ω → ι) : ι → Ω → β | λ i ω, u (min i (τ ω)) ω | def | measure_theory.stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | Given a map `u : ι → Ω → E`, the stopped process with respect to `τ` is `u i ω` if
`i ≤ τ ω`, and `u (τ ω) ω` otherwise.
Intuitively, the stopped process stops evolving once the stopping time has occured. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_process_eq_stopped_value {u : ι → Ω → β} {τ : Ω → ι} :
stopped_process u τ = λ i, stopped_value u (λ ω, min i (τ ω)) | rfl | lemma | measure_theory.stopped_process_eq_stopped_value | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_stopped_process {u : ι → Ω → β} {τ σ : Ω → ι} :
stopped_value (stopped_process u τ) σ = stopped_value u (λ ω, min (σ ω) (τ ω)) | rfl | lemma | measure_theory.stopped_value_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq_of_le {u : ι → Ω → β} {τ : Ω → ι}
{i : ι} {ω : Ω} (h : i ≤ τ ω) : stopped_process u τ i ω = u i ω | by simp [stopped_process, min_eq_left h] | lemma | measure_theory.stopped_process_eq_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq_of_ge {u : ι → Ω → β} {τ : Ω → ι}
{i : ι} {ω : Ω} (h : τ ω ≤ i) : stopped_process u τ i ω = u (τ ω) ω | by simp [stopped_process, min_eq_right h] | lemma | measure_theory.stopped_process_eq_of_ge | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable_min_stopping_time [metrizable_space ι] (hτ : is_stopping_time f τ) :
prog_measurable f (λ i ω, min i (τ ω)) | begin
intro i,
let m_prod : measurable_space (set.Iic i × Ω) := measurable_space.prod _ (f i),
let m_set : ∀ t : set (set.Iic i × Ω), measurable_space t :=
λ _, @subtype.measurable_space (set.Iic i × Ω) _ m_prod,
let s := {p : set.Iic i × Ω | τ p.2 ≤ i},
have hs : measurable_set[m_prod] s, from @measurabl... | lemma | measure_theory.prog_measurable_min_stopping_time | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"iff_and_self",
"le_min_iff",
"measurable",
"measurable.min",
"measurable.strongly_measurable",
"measurable_of_Iic",
"measurable_of_restrict_of_restrict_compl",
"measurable_set",
"measurable_snd",
"measurable_space",
"measurable_space.prod",
"measurable_subtype_coe",
"set.Iic",
"set.mem_Ii... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable.stopped_process [metrizable_space ι]
(h : prog_measurable f u) (hτ : is_stopping_time f τ) :
prog_measurable f (stopped_process u τ) | h.comp (prog_measurable_min_stopping_time hτ) (λ i x, min_le_left _ _) | lemma | measure_theory.prog_measurable.stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable.adapted_stopped_process [metrizable_space ι]
(h : prog_measurable f u) (hτ : is_stopping_time f τ) :
adapted f (stopped_process u τ) | (h.stopped_process hτ).adapted | lemma | measure_theory.prog_measurable.adapted_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prog_measurable.strongly_measurable_stopped_process [metrizable_space ι]
(hu : prog_measurable f u) (hτ : is_stopping_time f τ) (i : ι) :
strongly_measurable (stopped_process u τ i) | (hu.adapted_stopped_process hτ i).mono (f.le _) | lemma | measure_theory.prog_measurable.strongly_measurable_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_stopped_value_of_le
(h : prog_measurable f u) (hτ : is_stopping_time f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) :
strongly_measurable[f n] (stopped_value u τ) | begin
have : stopped_value u τ = (λ (p : set.Iic n × Ω), u ↑(p.fst) p.snd) ∘ (λ ω, (⟨τ ω, hτ_le ω⟩, ω)),
{ ext1 ω, simp only [stopped_value, function.comp_app, subtype.coe_mk], },
rw this,
refine strongly_measurable.comp_measurable (h n) _,
exact (hτ.measurable_of_le hτ_le).subtype_mk.prod_mk measurable_id,
e... | lemma | measure_theory.strongly_measurable_stopped_value_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_id",
"set.Iic",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_stopped_value [metrizable_space β] [measurable_space β] [borel_space β]
(hf_prog : prog_measurable f u) (hτ : is_stopping_time f τ) :
measurable[hτ.measurable_space] (stopped_value u τ) | begin
have h_str_meas : ∀ i, strongly_measurable[f i] (stopped_value u (λ ω, min (τ ω) i)),
from λ i, strongly_measurable_stopped_value_of_le hf_prog (hτ.min_const i)
(λ _, min_le_right _ _),
intros t ht i,
suffices : stopped_value u τ ⁻¹' t ∩ {ω : Ω | τ ω ≤ i}
= stopped_value u (λ ω, min (τ ω) i)... | lemma | measure_theory.measurable_stopped_value | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_left_iff",
"borel_space",
"measurable",
"measurable_space",
"set.mem_inter_iff",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_eq_of_mem_finset [add_comm_monoid E] {s : finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
stopped_value u τ = ∑ i in s, set.indicator {ω | τ ω = i} (u i) | begin
ext y,
rw [stopped_value, finset.sum_apply, finset.sum_indicator_eq_sum_filter],
suffices : finset.filter (λ i, y ∈ {ω : Ω | τ ω = i}) s = ({τ y} : finset ι),
by rw [this, finset.sum_singleton],
ext1 ω,
simp only [set.mem_set_of_eq, finset.mem_filter, finset.mem_singleton],
split; intro h,
{ exa... | lemma | measure_theory.stopped_value_eq_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_monoid",
"finset",
"finset.filter",
"finset.mem_filter",
"finset.mem_singleton",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_eq' [preorder ι] [locally_finite_order_bot ι] [add_comm_monoid E]
{N : ι} (hbdd : ∀ ω, τ ω ≤ N) :
stopped_value u τ = ∑ i in finset.Iic N, set.indicator {ω | τ ω = i} (u i) | stopped_value_eq_of_mem_finset (λ ω, finset.mem_Iic.mpr (hbdd ω)) | lemma | measure_theory.stopped_value_eq' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_monoid",
"finset.Iic",
"locally_finite_order_bot",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq_of_mem_finset [linear_order ι] [add_comm_monoid E]
{s : finset ι} (n : ι) (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) :
stopped_process u τ n =
set.indicator {a | n ≤ τ a} (u n) + ∑ i in s.filter (< n), set.indicator {ω | τ ω = i} (u i) | begin
ext ω,
rw [pi.add_apply, finset.sum_apply],
cases le_or_lt n (τ ω),
{ rw [stopped_process_eq_of_le h, set.indicator_of_mem, finset.sum_eq_zero, add_zero],
{ intros m hm,
refine set.indicator_of_not_mem _ _,
rw [finset.mem_filter] at hm,
exact (hm.2.trans_le h).ne', },
{ exact h, ... | lemma | measure_theory.stopped_process_eq_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_monoid",
"finset",
"finset.mem_filter",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq'' [linear_order ι] [locally_finite_order_bot ι] [add_comm_monoid E]
(n : ι) :
stopped_process u τ n =
set.indicator {a | n ≤ τ a} (u n) + ∑ i in finset.Iio n, set.indicator {ω | τ ω = i} (u i) | begin
have h_mem : ∀ ω, τ ω < n → τ ω ∈ finset.Iio n := λ ω h, finset.mem_Iio.mpr h,
rw stopped_process_eq_of_mem_finset n h_mem,
swap, { apply_instance, },
congr' with i,
simp only [finset.Iio_filter_lt, min_eq_right],
end | lemma | measure_theory.stopped_process_eq'' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_monoid",
"finset.Iio",
"finset.Iio_filter_lt",
"locally_finite_order_bot",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℒp_stopped_value_of_mem_finset (hτ : is_stopping_time ℱ τ) (hu : ∀ n, mem_ℒp (u n) p μ)
{s : finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
mem_ℒp (stopped_value u τ) p μ | begin
rw stopped_value_eq_of_mem_finset hbdd,
swap, apply_instance,
refine mem_ℒp_finset_sum' _ (λ i hi, mem_ℒp.indicator _ (hu i)),
refine ℱ.le i {a : Ω | τ a = i} (hτ.measurable_set_eq_of_countable_range _ i),
refine ((finset.finite_to_set s).subset (λ ω hω, _)).countable,
obtain ⟨y, rfl⟩ := hω,
exact h... | lemma | measure_theory.mem_ℒp_stopped_value_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"finset",
"finset.finite_to_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℒp_stopped_value [locally_finite_order_bot ι]
(hτ : is_stopping_time ℱ τ) (hu : ∀ n, mem_ℒp (u n) p μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) :
mem_ℒp (stopped_value u τ) p μ | mem_ℒp_stopped_value_of_mem_finset hτ hu (λ ω, finset.mem_Iic.mpr (hbdd ω)) | lemma | measure_theory.mem_ℒp_stopped_value | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"locally_finite_order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_stopped_value_of_mem_finset (hτ : is_stopping_time ℱ τ)
(hu : ∀ n, integrable (u n) μ) {s : finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
integrable (stopped_value u τ) μ | begin
simp_rw ← mem_ℒp_one_iff_integrable at hu ⊢,
exact mem_ℒp_stopped_value_of_mem_finset hτ hu hbdd,
end | lemma | measure_theory.integrable_stopped_value_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_stopped_value [locally_finite_order_bot ι]
(hτ : is_stopping_time ℱ τ) (hu : ∀ n, integrable (u n) μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) :
integrable (stopped_value u τ) μ | integrable_stopped_value_of_mem_finset hτ hu (λ ω, finset.mem_Iic.mpr (hbdd ω)) | lemma | measure_theory.integrable_stopped_value | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"locally_finite_order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℒp_stopped_process_of_mem_finset (hτ : is_stopping_time ℱ τ)
(hu : ∀ n, mem_ℒp (u n) p μ) (n : ι) {s : finset ι} (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) :
mem_ℒp (stopped_process u τ n) p μ | begin
rw stopped_process_eq_of_mem_finset n hbdd,
swap, { apply_instance, },
refine mem_ℒp.add _ _,
{ exact mem_ℒp.indicator (ℱ.le n {a : Ω | n ≤ τ a} (hτ.measurable_set_ge n)) (hu n) },
{ suffices : mem_ℒp (λ ω, ∑ i in s.filter (< n), {a : Ω | τ a = i}.indicator (u i) ω) p μ,
{ convert this, ext1 ω, simp... | lemma | measure_theory.mem_ℒp_stopped_process_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℒp_stopped_process [locally_finite_order_bot ι] (hτ : is_stopping_time ℱ τ)
(hu : ∀ n, mem_ℒp (u n) p μ) (n : ι) :
mem_ℒp (stopped_process u τ n) p μ | mem_ℒp_stopped_process_of_mem_finset hτ hu n (λ ω h, finset.mem_Iio.mpr h) | lemma | measure_theory.mem_ℒp_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"locally_finite_order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_stopped_process_of_mem_finset (hτ : is_stopping_time ℱ τ)
(hu : ∀ n, integrable (u n) μ) (n : ι) {s : finset ι} (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) :
integrable (stopped_process u τ n) μ | begin
simp_rw ← mem_ℒp_one_iff_integrable at hu ⊢,
exact mem_ℒp_stopped_process_of_mem_finset hτ hu n hbdd,
end | lemma | measure_theory.integrable_stopped_process_of_mem_finset | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_stopped_process [locally_finite_order_bot ι] (hτ : is_stopping_time ℱ τ)
(hu : ∀ n, integrable (u n) μ) (n : ι) :
integrable (stopped_process u τ n) μ | integrable_stopped_process_of_mem_finset hτ hu n (λ ω h, finset.mem_Iio.mpr h) | lemma | measure_theory.integrable_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"locally_finite_order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.stopped_process [metrizable_space ι]
(hu : adapted f u) (hu_cont : ∀ ω, continuous (λ i, u i ω)) (hτ : is_stopping_time f τ) :
adapted f (stopped_process u τ) | ((hu.prog_measurable_of_continuous hu_cont).stopped_process hτ).adapted | lemma | measure_theory.adapted.stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"continuous"
] | The stopped process of an adapted process with continuous paths is adapted. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adapted.stopped_process_of_discrete [discrete_topology ι]
(hu : adapted f u) (hτ : is_stopping_time f τ) :
adapted f (stopped_process u τ) | (hu.prog_measurable_of_discrete.stopped_process hτ).adapted | lemma | measure_theory.adapted.stopped_process_of_discrete | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"discrete_topology"
] | If the indexing order has the discrete topology, then the stopped process of an adapted process
is adapted. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adapted.strongly_measurable_stopped_process [metrizable_space ι]
(hu : adapted f u) (hu_cont : ∀ ω, continuous (λ i, u i ω)) (hτ : is_stopping_time f τ)
(n : ι) :
strongly_measurable (stopped_process u τ n) | (hu.prog_measurable_of_continuous hu_cont).strongly_measurable_stopped_process hτ n | lemma | measure_theory.adapted.strongly_measurable_stopped_process | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.strongly_measurable_stopped_process_of_discrete [discrete_topology ι]
(hu : adapted f u) (hτ : is_stopping_time f τ) (n : ι) :
strongly_measurable (stopped_process u τ n) | hu.prog_measurable_of_discrete.strongly_measurable_stopped_process hτ n | lemma | measure_theory.adapted.strongly_measurable_stopped_process_of_discrete | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_sub_eq_sum [add_comm_group β] (hle : τ ≤ π) :
stopped_value u π - stopped_value u τ =
λ ω, (∑ i in finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω | begin
ext ω,
rw [finset.sum_Ico_eq_sub _ (hle ω), finset.sum_range_sub, finset.sum_range_sub],
simp [stopped_value],
end | lemma | measure_theory.stopped_value_sub_eq_sum | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_group",
"finset.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_sub_eq_sum' [add_comm_group β] (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ ω, π ω ≤ N) :
stopped_value u π - stopped_value u τ =
λ ω, (∑ i in finset.range (N + 1),
set.indicator {ω | τ ω ≤ i ∧ i < π ω} (u (i + 1) - u i)) ω | begin
rw stopped_value_sub_eq_sum hle,
ext ω,
simp only [finset.sum_apply, finset.sum_indicator_eq_sum_filter],
refine finset.sum_congr _ (λ _ _, rfl),
ext i,
simp only [finset.mem_filter, set.mem_set_of_eq, finset.mem_range, finset.mem_Ico],
exact ⟨λ h, ⟨lt_trans h.2 (nat.lt_succ_iff.2 $ hbdd _), h⟩, λ h... | lemma | measure_theory.stopped_value_sub_eq_sum' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_comm_group",
"finset.mem_Ico",
"finset.mem_filter",
"finset.mem_range",
"finset.range",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_eq {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) :
stopped_value u τ =
λ x, (∑ i in finset.range (N + 1), set.indicator {ω | τ ω = i} (u i)) x | stopped_value_eq_of_mem_finset (λ ω, finset.mem_range_succ_iff.mpr (hbdd ω)) | lemma | measure_theory.stopped_value_eq | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset.range",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq (n : ℕ) :
stopped_process u τ n =
set.indicator {a | n ≤ τ a} (u n) + ∑ i in finset.range n, set.indicator {ω | τ ω = i} (u i) | begin
rw stopped_process_eq'' n,
swap, { apply_instance, },
congr' with i,
rw [finset.mem_Iio, finset.mem_range],
end | lemma | measure_theory.stopped_process_eq | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset.mem_Iio",
"finset.mem_range",
"finset.range",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_process_eq' (n : ℕ) :
stopped_process u τ n =
set.indicator {a | n + 1 ≤ τ a} (u n) +
∑ i in finset.range (n + 1), set.indicator {a | τ a = i} (u i) | begin
have : {a | n ≤ τ a}.indicator (u n) =
{a | n + 1 ≤ τ a}.indicator (u n) + {a | τ a = n}.indicator (u n),
{ ext x,
rw [add_comm, pi.add_apply, ← set.indicator_union_of_not_mem_inter],
{ simp_rw [@eq_comm _ _ n, @le_iff_eq_or_lt _ _ n, nat.succ_le_iff],
refl },
{ rintro ⟨h₁, h₂⟩,
ex... | lemma | measure_theory.stopped_process_eq' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"finset.range",
"le_iff_eq_or_lt",
"nat.succ_le_iff",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.piecewise_of_le (hτ_st : is_stopping_time 𝒢 τ)
(hη_st : is_stopping_time 𝒢 η) (hτ : ∀ ω, i ≤ τ ω) (hη : ∀ ω, i ≤ η ω)
(hs : measurable_set[𝒢 i] s) :
is_stopping_time 𝒢 (s.piecewise τ η) | begin
intro n,
have : {ω | s.piecewise τ η ω ≤ n} = (s ∩ {ω | τ ω ≤ n}) ∪ (sᶜ ∩ {ω | η ω ≤ n}),
{ ext1 ω,
simp only [set.piecewise, set.mem_inter_iff, set.mem_set_of_eq, and.congr_right_iff],
by_cases hx : ω ∈ s; simp [hx], },
rw this,
by_cases hin : i ≤ n,
{ have hs_n : measurable_set[𝒢 n] s, from... | lemma | measure_theory.is_stopping_time.piecewise_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_right_iff",
"measurable_set",
"set.mem_inter_iff",
"set.piecewise"
] | Given stopping times `τ` and `η` which are bounded below, `set.piecewise s τ η` is also
a stopping time with respect to the same filtration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_stopping_time_piecewise_const (hij : i ≤ j) (hs : measurable_set[𝒢 i] s) :
is_stopping_time 𝒢 (s.piecewise (λ _, i) (λ _, j)) | (is_stopping_time_const 𝒢 i).piecewise_of_le (is_stopping_time_const 𝒢 j)
(λ x, le_rfl) (λ _, hij) hs | lemma | measure_theory.is_stopping_time_piecewise_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"le_rfl",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_piecewise_const {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} :
stopped_value f (s.piecewise (λ _, i) (λ _, j)) = s.piecewise (f i) (f j) | by { ext ω, rw stopped_value, by_cases hx : ω ∈ s; simp [hx] } | lemma | measure_theory.stopped_value_piecewise_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_piecewise_const' {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} :
stopped_value f (s.piecewise (λ _, i) (λ _, j)) = s.indicator (f i) + sᶜ.indicator (f j) | by { ext ω, rw stopped_value, by_cases hx : ω ∈ s; simp [hx] } | lemma | measure_theory.stopped_value_piecewise_const' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_stopping_time_ae_eq_restrict_eq_of_countable_range [sigma_finite_filtration μ ℱ]
(hτ : is_stopping_time ℱ τ) (h_countable : (set.range τ).countable)
[sigma_finite (μ.trim (hτ.measurable_space_le_of_countable_range h_countable))] (i : ι) :
μ[f | hτ.measurable_space] =ᵐ[μ.restrict {x | τ x = i}] μ[f | ℱ i] | begin
refine condexp_ae_eq_restrict_of_measurable_space_eq_on
(hτ.measurable_space_le_of_countable_range h_countable) (ℱ.le i)
(hτ.measurable_set_eq_of_countable_range' h_countable i) (λ t, _),
rw [set.inter_comm _ t, is_stopping_time.measurable_set_inter_eq_iff],
end | lemma | measure_theory.condexp_stopping_time_ae_eq_restrict_eq_of_countable_range | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"set.inter_comm",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_stopping_time_ae_eq_restrict_eq_of_countable [countable ι]
[sigma_finite_filtration μ ℱ]
(hτ : is_stopping_time ℱ τ) [sigma_finite (μ.trim hτ.measurable_space_le_of_countable)] (i : ι) :
μ[f | hτ.measurable_space] =ᵐ[μ.restrict {x | τ x = i}] μ[f | ℱ i] | condexp_stopping_time_ae_eq_restrict_eq_of_countable_range hτ (set.to_countable _) i | lemma | measure_theory.condexp_stopping_time_ae_eq_restrict_eq_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_min_stopping_time_ae_eq_restrict_le_const (hτ : is_stopping_time ℱ τ)
(i : ι) [sigma_finite (μ.trim (hτ.min_const i).measurable_space_le)] :
μ[f | (hτ.min_const i).measurable_space]
=ᵐ[μ.restrict {x | τ x ≤ i}] μ[f | hτ.measurable_space] | begin
haveI : sigma_finite (μ.trim hτ.measurable_space_le),
{ have h_le : (hτ.min_const i).measurable_space ≤ hτ.measurable_space,
{ rw is_stopping_time.measurable_space_min_const,
exact inf_le_left, },
exact sigma_finite_trim_mono _ h_le, },
refine (condexp_ae_eq_restrict_of_measurable_space_eq_on ... | lemma | measure_theory.condexp_min_stopping_time_ae_eq_restrict_le_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"inf_le_left",
"measurable_space",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_stopping_time_ae_eq_restrict_eq
[first_countable_topology ι] [sigma_finite_filtration μ ℱ]
(hτ : is_stopping_time ℱ τ) [sigma_finite (μ.trim hτ.measurable_space_le)] (i : ι) :
μ[f | hτ.measurable_space] =ᵐ[μ.restrict {x | τ x = i}] μ[f | ℱ i] | begin
refine condexp_ae_eq_restrict_of_measurable_space_eq_on
hτ.measurable_space_le (ℱ.le i) (hτ.measurable_set_eq' i) (λ t, _),
rw [set.inter_comm _ t, is_stopping_time.measurable_set_inter_eq_iff],
end | lemma | measure_theory.condexp_stopping_time_ae_eq_restrict_eq | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_min_stopping_time_ae_eq_restrict_le [measurable_space ι]
[second_countable_topology ι] [borel_space ι]
(hτ : is_stopping_time ℱ τ) (hσ : is_stopping_time ℱ σ)
[sigma_finite (μ.trim (hτ.min hσ).measurable_space_le)] :
μ[f | (hτ.min hσ).measurable_space] =ᵐ[μ.restrict {x | τ x ≤ σ x}] μ[f | hτ.measurable_... | begin
haveI : sigma_finite (μ.trim hτ.measurable_space_le),
{ have h_le : (hτ.min hσ).measurable_space ≤ hτ.measurable_space,
{ rw is_stopping_time.measurable_space_min,
exact inf_le_left, },
exact sigma_finite_trim_mono _ h_le, },
refine (condexp_ae_eq_restrict_of_measurable_space_eq_on hτ.measurab... | lemma | measure_theory.condexp_min_stopping_time_ae_eq_restrict_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"inf_le_left",
"measurable_space",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Action (G : Mon.{u}) | (V : V)
(ρ : G ⟶ Mon.of (End V)) | structure | Action | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Mon.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ρ_one {G : Mon.{u}} (A : Action V G) : A.ρ 1 = 𝟙 A.V | by { rw [monoid_hom.map_one], refl, } | lemma | Action.ρ_one | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"monoid_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ρ_Aut {G : Group.{u}} (A : Action V (Mon.of G)) : G ⟶ Group.of (Aut A.V) | { to_fun := λ g,
{ hom := A.ρ g,
inv := A.ρ (g⁻¹ : G),
hom_inv_id' := ((A.ρ).map_mul (g⁻¹ : G) g).symm.trans (by rw [inv_mul_self, ρ_one]),
inv_hom_id' := ((A.ρ).map_mul g (g⁻¹ : G)).symm.trans (by rw [mul_inv_self, ρ_one]), },
map_one' := by { ext, exact A.ρ.map_one },
map_mul' := λ x y, by { ext, ex... | def | Action.ρ_Aut | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Group.of",
"Mon.of",
"inv_mul_self",
"map_mul",
"mul_inv_self"
] | When a group acts, we can lift the action to the group of automorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited' : inhabited (Action (Type u) G) | ⟨⟨punit, 1⟩⟩ | instance | Action.inhabited' | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivial : Action AddCommGroup G | { V := AddCommGroup.of punit,
ρ := 1, } | def | Action.trivial | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | The trivial representation of a group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (M N : Action V G) | (hom : M.V ⟶ N.V)
(comm' : ∀ g : G, M.ρ g ≫ hom = hom ≫ N.ρ g . obviously) | structure | Action.hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | A homomorphism of `Action V G`s is a morphism between the underlying objects,
commuting with the action of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (M : Action V G) : Action.hom M M | { hom := 𝟙 M.V } | def | Action.hom.id | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.hom"
] | The identity morphism on a `Action V G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {M N K : Action V G} (p : Action.hom M N) (q : Action.hom N K) :
Action.hom M K | { hom := p.hom ≫ q.hom,
comm' := λ g, by rw [←category.assoc, p.comm, category.assoc, q.comm, ←category.assoc] } | def | Action.hom.comp | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.hom"
] | The composition of two `Action V G` homomorphisms is the composition of the underlying maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_hom (M : Action V G) : (𝟙 M : hom M M).hom = 𝟙 M.V | rfl | lemma | Action.id_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :
(f ≫ g : hom M K).hom = f.hom ≫ g.hom | rfl | lemma | Action.comp_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_iso {M N : Action V G} (f : M.V ≅ N.V) (comm : ∀ g : G, M.ρ g ≫ f.hom = f.hom ≫ N.ρ g) :
M ≅ N | { hom :=
{ hom := f.hom,
comm' := comm, },
inv :=
{ hom := f.inv,
comm' := λ g, by { have w := comm g =≫ f.inv, simp at w, simp [w], }, }} | def | Action.mk_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"comm"
] | Construct an isomorphism of `G` actions/representations
from an isomorphism of the the underlying objects,
where the forward direction commutes with the group action. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_hom_is_iso {M N : Action V G} (f : M ⟶ N) [is_iso f.hom] : is_iso f | by { convert is_iso.of_iso (mk_iso (as_iso f.hom) f.comm), ext, refl, } | instance | Action.is_iso_of_hom_is_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_hom_mk {M N : Action V G} (f : M.V ⟶ N.V) [is_iso f] (w) :
@is_iso _ _ M N ⟨f, w⟩ | is_iso.of_iso (mk_iso (as_iso f) w) | instance | Action.is_iso_hom_mk | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor : Action V G ⥤ (single_obj G ⥤ V) | { obj := λ M,
{ obj := λ _, M.V,
map := λ _ _ g, M.ρ g,
map_id' := λ _, M.ρ.map_one,
map_comp' := λ _ _ _ g h, M.ρ.map_mul h g, },
map := λ M N f,
{ app := λ _, f.hom,
naturality' := λ _ _ g, f.comm g, } } | def | Action.functor_category_equivalence.functor | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | Auxilliary definition for `functor_category_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse : (single_obj G ⥤ V) ⥤ Action V G | { obj := λ F,
{ V := F.obj punit.star,
ρ :=
{ to_fun := λ g, F.map g,
map_one' := F.map_id punit.star,
map_mul' := λ g h, F.map_comp h g, } },
map := λ M N f,
{ hom := f.app punit.star,
comm' := λ g, f.naturality g, } }. | def | Action.functor_category_equivalence.inverse | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | Auxilliary definition for `functor_category_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso : 𝟭 (Action V G) ≅ functor ⋙ inverse | nat_iso.of_components (λ M, mk_iso ((iso.refl _)) (by tidy)) (by tidy). | def | Action.functor_category_equivalence.unit_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | Auxilliary definition for `functor_category_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_iso : inverse ⋙ functor ≅ 𝟭 (single_obj G ⥤ V) | nat_iso.of_components (λ M, nat_iso.of_components (by tidy) (by tidy)) (by tidy). | def | Action.functor_category_equivalence.counit_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | Auxilliary definition for `functor_category_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_equivalence : Action V G ≌ (single_obj G ⥤ V) | { functor := functor,
inverse := inverse,
unit_iso := unit_iso,
counit_iso := counit_iso, } | def | Action.functor_category_equivalence | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | The category of actions of `G` in the category `V`
is equivalent to the functor category `single_obj G ⥤ V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_equivalence.functor_def :
(functor_category_equivalence V G).functor = functor_category_equivalence.functor | rfl | lemma | Action.functor_category_equivalence.functor_def | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_equivalence.inverse_def :
(functor_category_equivalence V G).inverse = functor_category_equivalence.inverse | rfl | lemma | Action.functor_category_equivalence.inverse_def | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : Action V G ⥤ V | { obj := λ M, M.V,
map := λ M N f, f.hom, } | def | Action.forget | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | (implementation) The forgetful functor from bundled actions to the underlying objects.
Use the `category_theory.forget` API provided by the `concrete_category` instance below,
rather than using this directly. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_V [concrete_category V] : has_forget₂ (Action V G) V | { forget₂ := forget V G } | instance | Action.has_forget_to_V | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_equivalence_comp_evaluation :
(functor_category_equivalence V G).functor ⋙ (evaluation _ _).obj punit.star ≅ forget V G | iso.refl _ | def | Action.functor_category_equivalence_comp_evaluation | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | The forgetful functor is intertwined by `functor_category_equivalence` with
evaluation at `punit.star`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso.conj_ρ {M N : Action V G} (f : M ≅ N) (g : G) :
N.ρ g = (((forget V G).map_iso f).conj (M.ρ g)) | by { rw [iso.conj_apply, iso.eq_inv_comp], simp [f.hom.comm'] } | lemma | Action.iso.conj_ρ | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_zero_morphisms : functor.preserves_zero_morphisms (forget V G) | {} | instance | Action.forget_preserves_zero_morphisms | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_preserves_zero_morphisms [concrete_category V] :
functor.preserves_zero_morphisms (forget₂ (Action V G) V) | {} | instance | Action.forget₂_preserves_zero_morphisms | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_equivalence_preserves_zero_morphisms :
functor.preserves_zero_morphisms (functor_category_equivalence V G).functor | {} | instance | Action.functor_category_equivalence_preserves_zero_morphisms | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_additive :
functor.additive (forget V G) | {} | instance | Action.forget_additive | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_additive [concrete_category V] :
functor.additive (forget₂ (Action V G) V) | {} | instance | Action.forget₂_additive | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_equivalence_additive :
functor.additive (functor_category_equivalence V G).functor | {} | instance | Action.functor_category_equivalence_additive | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_hom {X Y : Action V G} : (0 : X ⟶ Y).hom = 0 | rfl | lemma | Action.zero_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_hom {X Y : Action V G} (f : X ⟶ Y) : (-f).hom = -f.hom | rfl | lemma | Action.neg_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_hom {X Y : Action V G} (f g : X ⟶ Y) : (f + g).hom = f.hom + g.hom | rfl | lemma | Action.add_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_hom {X Y : Action V G} {ι : Type*} (f : ι → (X ⟶ Y)) (s : finset ι) :
(s.sum f).hom = s.sum (λ i, (f i).hom) | (forget V G).map_sum f s | lemma | Action.sum_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_linear :
functor.linear R (forget V G) | {} | instance | Action.forget_linear | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_linear [concrete_category V] :
functor.linear R (forget₂ (Action V G) V) | {} | instance | Action.forget₂_linear | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_equivalence_linear :
functor.linear R (functor_category_equivalence V G).functor | {} | instance | Action.functor_category_equivalence_linear | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_hom {X Y : Action V G} (r : R) (f : X ⟶ Y) : (r • f).hom = r • f.hom | rfl | lemma | Action.smul_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_aux : Action V G ≌ (ulift.{u} (single_obj G) ⥤ V) | (functor_category_equivalence V G).trans (equivalence.congr_left ulift.equivalence) | def | Action.abelian_aux | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | Auxilliary construction for the `abelian (Action V G)` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_unit_V : (𝟙_ (Action V G)).V = 𝟙_ V | rfl | lemma | Action.tensor_unit_V | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_unit_rho {g : G} : (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) | rfl | lemma | Action.tensor_unit_rho | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_V {X Y : Action V G} : (X ⊗ Y).V = X.V ⊗ Y.V | rfl | lemma | Action.tensor_V | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_rho {X Y : Action V G} {g : G} : (X ⊗ Y).ρ g = X.ρ g ⊗ Y.ρ g | rfl | lemma | Action.tensor_rho | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_hom {W X Y Z : Action V G} (f : W ⟶ X) (g : Y ⟶ Z) :
(f ⊗ g).hom = f.hom ⊗ g.hom | rfl | lemma | Action.tensor_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_hom_hom {X Y Z : Action V G} :
hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom | begin
dsimp [monoidal.transport_associator],
simp,
end | lemma | Action.associator_hom_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_inv_hom {X Y Z : Action V G} :
hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv | begin
dsimp [monoidal.transport_associator],
simp,
end | lemma | Action.associator_inv_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_hom_hom {X : Action V G} :
hom.hom (λ_ X).hom = (λ_ X.V).hom | begin
dsimp [monoidal.transport_left_unitor],
simp,
end | lemma | Action.left_unitor_hom_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_inv_hom {X : Action V G} :
hom.hom (λ_ X).inv = (λ_ X.V).inv | begin
dsimp [monoidal.transport_left_unitor],
simp,
end | lemma | Action.left_unitor_inv_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_hom_hom {X : Action V G} :
hom.hom (ρ_ X).hom = (ρ_ X.V).hom | begin
dsimp [monoidal.transport_right_unitor],
simp,
end | lemma | Action.right_unitor_hom_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_inv_hom {X : Action V G} :
hom.hom (ρ_ X).inv = (ρ_ X.V).inv | begin
dsimp [monoidal.transport_right_unitor],
simp,
end | lemma | Action.right_unitor_inv_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_unit_iso {X : V} (f : 𝟙_ V ≅ X) :
𝟙_ (Action V G) ≅ Action.mk X 1 | Action.mk_iso f (λ g, by simp only [monoid_hom.one_apply, End.one_def, category.id_comp f.hom,
tensor_unit_rho, category.comp_id]) | def | Action.tensor_unit_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.mk_iso",
"monoid_hom.one_apply"
] | Given an object `X` isomorphic to the tensor unit of `V`, `X` equipped with the trivial action
is isomorphic to the tensor unit of `Action V G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_monoidal : monoidal_functor (Action V G) V | { ε := 𝟙 _,
μ := λ X Y, 𝟙 _,
..Action.forget _ _, } | def | Action.forget_monoidal | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.forget"
] | When `V` is monoidal the forgetful functor `Action V G` to `V` is monoidal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_monoidal_faithful : faithful (forget_monoidal V G).to_functor | by { change faithful (forget V G), apply_instance, } | instance | Action.forget_monoidal_faithful | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_braided : braided_functor (Action V G) V | { ..forget_monoidal _ _, } | def | Action.forget_braided | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | When `V` is braided the forgetful functor `Action V G` to `V` is braided. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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