# Theory Convolution

theory Convolution
imports Independent_Family
(*  Title:      HOL/Probability/Convolution.thy
Author:     Sudeep Kanav, TU München
Author:     Johannes Hölzl, TU München *)

section ‹Convolution Measure›

theory Convolution
imports Independent_Family
begin

lemma (in finite_measure) sigma_finite_measure: "sigma_finite_measure M"
..

definition convolution :: "('a :: ordered_euclidean_space) measure ⇒ 'a measure ⇒ 'a measure" (infix "⋆" 50) where
"convolution M N = distr (M ⨂M N) borel (λ(x, y). x + y)"

lemma
shows space_convolution[simp]: "space (convolution M N) = space borel"
and sets_convolution[simp]: "sets (convolution M N) = sets borel"
and measurable_convolution1[simp]: "measurable A (convolution M N) = measurable A borel"
and measurable_convolution2[simp]: "measurable (convolution M N) B = measurable borel B"

lemma nn_integral_convolution:
assumes "finite_measure M" "finite_measure N"
assumes [measurable_cong]: "sets N = sets borel" "sets M = sets borel"
assumes [measurable]: "f ∈ borel_measurable borel"
shows "(∫+x. f x ∂convolution M N) = (∫+x. ∫+y. f (x + y) ∂N ∂M)"
proof -
interpret M: finite_measure M by fact
interpret N: finite_measure N by fact
interpret pair_sigma_finite M N ..
show ?thesis
unfolding convolution_def
qed

lemma convolution_emeasure:
assumes "A ∈ sets borel" "finite_measure M" "finite_measure N"
assumes [simp]: "sets N = sets borel" "sets M = sets borel"
assumes [simp]: "space M = space N" "space N = space borel"
shows "emeasure (M ⋆ N) A = ∫+x. (emeasure N {a. a + x ∈ A}) ∂M "
using assms by (auto intro!: nn_integral_cong simp del: nn_integral_indicator simp: nn_integral_convolution
nn_integral_indicator [symmetric] ac_simps split:split_indicator)

lemma convolution_emeasure':
assumes [simp]:"A ∈ sets borel"
assumes [simp]: "finite_measure M" "finite_measure N"
assumes [simp]: "sets N = sets borel" "sets M = sets borel"
shows  "emeasure (M ⋆ N) A = ∫+x. ∫+y.  (indicator  A (x + y)) ∂N  ∂M"
by (auto simp del: nn_integral_indicator simp: nn_integral_convolution
nn_integral_indicator[symmetric] borel_measurable_indicator)

lemma convolution_finite:
assumes [simp]: "finite_measure M" "finite_measure N"
assumes [measurable_cong]: "sets N = sets borel" "sets M = sets borel"
shows "finite_measure (M ⋆ N)"
unfolding convolution_def
by (intro finite_measure_pair_measure finite_measure.finite_measure_distr) auto

lemma convolution_emeasure_3:
assumes [simp, measurable]: "A ∈ sets borel"
assumes [simp]: "finite_measure M" "finite_measure N" "finite_measure L"
assumes [simp]: "sets N = sets borel" "sets M = sets borel" "sets L = sets borel"
shows "emeasure (L ⋆ (M ⋆ N )) A = ∫+x. ∫+y. ∫+z. indicator A (x + y + z) ∂N ∂M ∂L"
apply (subst nn_integral_indicator[symmetric], simp)
apply (subst nn_integral_convolution,
auto intro!: borel_measurable_indicator borel_measurable_indicator' convolution_finite)+

lemma convolution_emeasure_3':
assumes [simp, measurable]:"A ∈ sets borel"
assumes [simp]: "finite_measure M" "finite_measure N"  "finite_measure L"
assumes [measurable_cong, simp]: "sets N = sets borel" "sets M = sets borel" "sets L = sets borel"
shows "emeasure ((L ⋆ M) ⋆ N ) A = ∫+x. ∫+y. ∫+z. indicator A (x + y + z) ∂N ∂M ∂L"
apply (subst nn_integral_indicator[symmetric], simp)+
apply (subst nn_integral_convolution)
apply (subst nn_integral_convolution)
done

lemma convolution_commutative:
assumes [simp]: "finite_measure M" "finite_measure N"
assumes [measurable_cong, simp]: "sets N = sets borel" "sets M = sets borel"
shows "(M ⋆ N) = (N ⋆ M)"
proof (rule measure_eqI)
interpret M: finite_measure M by fact
interpret N: finite_measure N by fact
interpret pair_sigma_finite M N ..

show "sets (M ⋆ N) = sets (N ⋆ M)" by simp

fix A assume "A ∈ sets (M ⋆ N)"
then have 1[measurable]:"A ∈ sets borel" by simp
have "emeasure (M ⋆ N) A = ∫+x. ∫+y. indicator A (x + y) ∂N ∂M" by (auto intro!: convolution_emeasure')
also have "... = ∫+x. ∫+y. (λ(x,y). indicator A (x + y)) (x, y) ∂N ∂M" by (auto intro!: nn_integral_cong)
also have "... = ∫+y. ∫+x. (λ(x,y). indicator A (x + y)) (x, y) ∂M ∂N" by (rule Fubini[symmetric]) simp
also have "... = emeasure (N ⋆ M) A" by (auto intro!: nn_integral_cong simp: add.commute convolution_emeasure')
finally show "emeasure (M ⋆ N) A = emeasure (N ⋆ M) A" by simp
qed

lemma convolution_associative:
assumes [simp]: "finite_measure M" "finite_measure N"  "finite_measure L"
assumes [simp]: "sets N = sets borel" "sets M = sets borel" "sets L = sets borel"
shows "(L ⋆ (M ⋆ N)) = ((L ⋆ M) ⋆ N)"
by (auto intro!: measure_eqI simp: convolution_emeasure_3 convolution_emeasure_3')

lemma (in prob_space) sum_indep_random_variable:
assumes ind: "indep_var borel X borel Y"
assumes [simp, measurable]: "random_variable borel X"
assumes [simp, measurable]: "random_variable borel Y"
shows "distr M borel (λx. X x + Y x) = convolution (distr M borel X)  (distr M borel Y)"
using ind unfolding indep_var_distribution_eq convolution_def
by (auto simp: distr_distr intro!:arg_cong[where f = "distr M borel"])

lemma (in prob_space) sum_indep_random_variable_lborel:
assumes ind: "indep_var borel X borel Y"
assumes [simp, measurable]: "random_variable lborel X"
assumes [simp, measurable]:"random_variable lborel Y"
shows "distr M lborel (λx. X x + Y x) = convolution (distr M lborel X)  (distr M lborel Y)"
using ind unfolding indep_var_distribution_eq convolution_def
by (auto simp: distr_distr o_def intro!: arg_cong[where f = "distr M borel"] cong: distr_cong)

lemma convolution_density:
fixes f g :: "real ⇒ ennreal"
assumes [measurable]: "f ∈ borel_measurable borel" "g ∈ borel_measurable borel"
assumes [simp]:"finite_measure (density lborel f)" "finite_measure (density lborel g)"
shows "density lborel f ⋆ density lborel g = density lborel (λx. ∫+y. f (x - y) * g y ∂lborel)"
(is "?l = ?r")
proof (intro measure_eqI)
fix A assume "A ∈ sets ?l"
then have [measurable]: "A ∈ sets borel"
by simp

have "(∫+x. f x * (∫+y. g y * indicator A (x + y) ∂lborel) ∂lborel) =
(∫+x. (∫+y. g y * (f x * indicator A (x + y)) ∂lborel) ∂lborel)"
proof (intro nn_integral_cong_AE, eventually_elim)
fix x
have "f x * (∫+ y. g y * indicator A (x + y) ∂lborel) =
(∫+ y. f x * (g y * indicator A (x + y)) ∂lborel)"
by (intro nn_integral_cmult[symmetric]) auto
then show "f x * (∫+ y. g y * indicator A (x + y) ∂lborel) =
(∫+ y. g y * (f x * indicator A (x + y)) ∂lborel)"
qed
also have "… = (∫+y. (∫+x. g y * (f x * indicator A (x + y)) ∂lborel) ∂lborel)"
by (intro lborel_pair.Fubini') simp
also have "… = (∫+y. (∫+x. f (x - y) * g y * indicator A x ∂lborel) ∂lborel)"
proof (intro nn_integral_cong_AE, eventually_elim)
fix y
have "(∫+x. g y * (f x * indicator A (x + y)) ∂lborel) =
g y * (∫+x. f x * indicator A (x + y) ∂lborel)"
by (intro nn_integral_cmult) auto
also have "… = g y * (∫+x. f (x - y) * indicator A x ∂lborel)"
by (subst nn_integral_real_affine[where c=1 and t="-y"])
also have "… = (∫+x. g y * (f (x - y) * indicator A x) ∂lborel)"
by (intro nn_integral_cmult[symmetric]) auto
finally show "(∫+ x. g y * (f x * indicator A (x + y)) ∂lborel) =
(∫+ x. f (x - y) * g y * indicator A x ∂lborel)"
qed
also have "… = (∫+x. (∫+y. f (x - y) * g y * indicator A x ∂lborel) ∂lborel)"
by (intro lborel_pair.Fubini') simp
finally show "emeasure ?l A = emeasure ?r A"
by (auto simp: convolution_emeasure' nn_integral_density emeasure_density
nn_integral_multc)
qed simp

lemma (in prob_space) distributed_finite_measure_density:
"distributed M N X f ⟹ finite_measure (density N f)"
using finite_measure_distr[of X N] distributed_distr_eq_density[of M N X f] by simp

lemma (in prob_space) distributed_convolution:
fixes f :: "real ⇒ _"
fixes g :: "real ⇒ _"
assumes indep: "indep_var borel X borel Y"
assumes X: "distributed M lborel X f"
assumes Y: "distributed M lborel Y g"
shows "distributed M lborel (λx. X x + Y x) (λx. ∫+y. f (x - y) * g y ∂lborel)"
unfolding distributed_def
proof safe
have fg[measurable]: "f ∈ borel_measurable borel" "g ∈ borel_measurable borel"
using distributed_borel_measurable[OF X] distributed_borel_measurable[OF Y] by simp_all

show "(λx. ∫+ xa. f (x - xa) * g xa ∂lborel) ∈ borel_measurable lborel"
by measurable

have "distr M borel (λx. X x + Y x) = (distr M borel X ⋆ distr M borel Y)"
using distributed_measurable[OF X] distributed_measurable[OF Y]
by (intro sum_indep_random_variable) (auto simp: indep)
also have "… = (density lborel f ⋆ density lborel g)"
using distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
by (simp cong: distr_cong)
also have "… = density lborel (λx. ∫+ y. f (x - y) * g y ∂lborel)"
proof (rule convolution_density)
show "finite_measure (density lborel f)"
using X by (rule distributed_finite_measure_density)
show "finite_measure (density lborel g)"
using Y by (rule distributed_finite_measure_density)
qed fact+
finally show "distr M lborel (λx. X x + Y x) = density lborel (λx. ∫+ y. f (x - y) * g y ∂lborel)"
by (simp cong: distr_cong)
show "random_variable lborel (λx. X x + Y x)"
using distributed_measurable[OF X] distributed_measurable[OF Y] by simp
qed

lemma prob_space_convolution_density:
fixes f:: "real ⇒ _"
fixes g:: "real ⇒ _"
assumes [measurable]: "f∈ borel_measurable borel"
assumes [measurable]: "g∈ borel_measurable borel"
assumes gt_0[simp]: "⋀x. 0 ≤ f x" "⋀x. 0 ≤ g x"
assumes "prob_space (density lborel f)" (is "prob_space ?F")
assumes "prob_space (density lborel g)" (is "prob_space ?G")
shows "prob_space (density lborel (λx.∫+y. f (x - y) * g y ∂lborel))" (is "prob_space ?D")
proof (subst convolution_density[symmetric])
interpret F: prob_space ?F by fact
show "finite_measure ?F" by unfold_locales
interpret G: prob_space ?G by fact
show "finite_measure ?G" by unfold_locales
interpret FG: pair_prob_space ?F ?G ..

show "prob_space (density lborel f ⋆ density lborel g)"
unfolding convolution_def by (rule FG.prob_space_distr) simp
qed simp_all

end