| import OAI.Analysis.InterpolatedFactors.Mixed |
|
|
| namespace OAI |
|
|
| universe u_1 u_2 u_3 u_4 u_5 |
|
|
| noncomputable section |
| namespace FreeGroupFactorMain.RectangularAbsorption |
|
|
| section |
| variable {A : Type u_1} [Ring A] |
|
|
| def part (p q : A) (C : ℕ → A) (n : ℕ) : A := p*C n*q |
| def remainder (p q : A) (C : ℕ → A) (n : ℕ) : A := C n-part p q C n |
| def shifted (p q D : A) (C : ℕ → A) : ℕ → A |
| | 0 => D+remainder p q C 0 |
| | n+1 => part p q C n+remainder p q C (n+1) |
|
|
| theorem part_supported (p q : A) (hp : p*p=p) (hq : q*q=q) (C : ℕ → A) (n : ℕ) : |
| p*part p q C n*q=part p q C n := by |
| calc |
| _ = (p*p)*C n*(q*q) := by dsimp [part];noncomm_ring |
| _ = _ := by rw [hp,hq];rfl |
|
|
| theorem remainder_compression (p q : A) (hp : p*p=p) (hq : q*q=q) (C : ℕ → A) (n : ℕ) : |
| p*remainder p q C n*q=0 := by |
| rw [remainder,mul_sub,sub_mul,part_supported p q hp hq] |
| exact sub_self _ |
|
|
| theorem shifted_zero_compression (p q D : A) (hp : p*p=p) (hq : q*q=q) |
| (hD : p*D*q=D) (C : ℕ → A) : p*shifted p q D C 0*q=D := by |
| rw [shifted,mul_add,add_mul,hD,remainder_compression p q hp hq,add_zero] |
|
|
| theorem shifted_succ_compression (p q D : A) (hp : p*p=p) (hq : q*q=q) |
| (C : ℕ → A) (n : ℕ) : p*shifted p q D C (n+1)*q=part p q C n := by |
| rw [shifted,mul_add,add_mul,part_supported p q hp hq,remainder_compression p q hp hq,add_zero] |
|
|
| theorem recover (p q D : A) (hp : p*p=p) (hq : q*q=q) (hD : p*D*q=D) |
| (C : ℕ → A) (n : ℕ) : |
| C n=p*shifted p q D C (n+1)*q+shifted p q D C n-p*shifted p q D C n*q := by |
| rw [shifted_succ_compression p q D hp hq] |
| cases n with |
| | zero => rw [shifted_zero_compression p q D hp hq hD,shifted,remainder];abel |
| | succ n => rw [shifted_succ_compression p q D hp hq,shifted,remainder];abel |
|
|
| variable [StarRing A] |
| def paired (x y : A) : A := x+star y |
|
|
| theorem paired_compression (p q x y : A) (hp : star p=p) (hq : star q=q) : |
| paired (p*x*q) (q*y*p)=p*paired x y*q := by |
| simp only [paired,star_mul,hp,hq,mul_add,add_mul,mul_assoc] |
|
|
| theorem paired_shifted_zero (p q D E : A) (hp : star p=p) (hq : star q=q) |
| (C F : ℕ → A) : |
| paired (shifted p q D C 0) (shifted q p E F 0)= |
| paired D E+paired (C 0) (F 0)-p*paired (C 0) (F 0)*q := by |
| simp only [paired,shifted,remainder,part,star_add,star_sub,star_mul,hp,hq] |
| noncomm_ring |
|
|
| theorem paired_shifted_succ (p q D E : A) (hp : star p=p) (hq : star q=q) |
| (C F : ℕ → A) (n : ℕ) : |
| paired (shifted p q D C (n+1)) (shifted q p E F (n+1))= |
| p*paired (C n) (F n)*q+paired (C (n+1)) (F (n+1))- |
| p*paired (C (n+1)) (F (n+1))*q := by |
| simp only [paired,shifted,remainder,part,star_add,star_sub,star_mul,hp,hq] |
| noncomm_ring |
|
|
| theorem recover_paired (p q D E : A) (hp : p*p=p) (hq : q*q=q) |
| (hps : star p=p) (hqs : star q=q) (hD : p*D*q=D) (hE : q*E*p=E) |
| (C F : ℕ → A) (n : ℕ) : |
| paired (C n) (F n)= |
| p*paired (shifted p q D C (n+1)) (shifted q p E F (n+1))*q+ |
| paired (shifted p q D C n) (shifted q p E F n)- |
| p*paired (shifted p q D C n) (shifted q p E F n)*q := by |
| have h := congrArg₂ paired (recover p q D hp hq hD C n) (recover q p E hq hp hE F n) |
| rw [h] |
| simp only [paired,star_sub,star_add,star_mul,hps,hqs] |
| noncomm_ring |
|
|
| theorem recover_supported_paired (p q D E : A) (hp : p*p=p) (hq : q*q=q) |
| (hps : star p=p) (hqs : star q=q) (hD : p*D*q=D) (hE : q*E*p=E) |
| (C F : ℕ → A) : |
| paired D E=p*paired (shifted p q D C 0) (shifted q p E F 0)*q := by |
| rw [← paired_compression p q _ _ hps hqs, |
| shifted_zero_compression p q D hp hq hD,shifted_zero_compression q p E hq hp hE] |
|
|
| variable [Algebra ℂ A] [StarModule ℂ A] |
|
|
| theorem generation_equivalence (p q D E : A) (hp : p*p=p) (hq : q*q=q) |
| (hps : star p=p) (hqs : star q=q) (hD : p*D*q=D) (hE : q*E*p=E) |
| (C F : ℕ → A) (M : StarSubalgebra ℂ A) (hMp : p∈M) (hMq : q∈M) : |
| (paired D E∈M ∧ ∀ n,paired (C n) (F n)∈M) ↔ |
| ∀ n,paired (shifted p q D C n) (shifted q p E F n)∈M := by |
| constructor |
| · rintro ⟨hDE,hCF⟩ n |
| cases n with |
| | zero => |
| rw [paired_shifted_zero p q D E hps hqs] |
| exact sub_mem (add_mem hDE (hCF 0)) (mul_mem (mul_mem hMp (hCF 0)) hMq) |
| | succ n => |
| rw [paired_shifted_succ p q D E hps hqs] |
| exact sub_mem (add_mem (mul_mem (mul_mem hMp (hCF n)) hMq) (hCF (n+1))) |
| (mul_mem (mul_mem hMp (hCF (n+1))) hMq) |
| · intro h |
| constructor |
| · rw [recover_supported_paired p q D E hp hq hps hqs hD hE C F] |
| exact mul_mem (mul_mem hMp (h 0)) hMq |
| · intro n |
| rw [recover_paired p q D E hp hq hps hqs hD hE C F] |
| exact sub_mem (add_mem (mul_mem (mul_mem hMp (h (n+1))) hMq) (h n)) |
| (mul_mem (mul_mem hMp (h n)) hMq) |
|
|
| end |
| section |
| variable {A : Type u_2} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A] |
|
|
| structure Data (B : StarSubalgebra ℂ A) where |
| p : B |
| q : B |
| projection : IsStarProjection p |
| projectionq : IsStarProjection q |
| τ : B →ₗ[ℂ] ℂ |
| tracial : ∀ a b,τ (a*b)=τ (b*a) |
| C : ℕ → A |
| D : A |
| D_supported : (p:A)*D*(q:A)=D |
| C_cov : ∀ (a : B) n m,star (C n)*(a:A)*C m=if n=m then τ a • (1:A) else 0 |
| D_cov : ∀ (a : B),star D*(a:A)*D=τ (a*p) • (q:A) |
| C_D : ∀ (a : B) n,star (C n)*(a:A)*D=0 |
| D_C : ∀ (a : B) n,star D*(a:A)*C n=0 |
|
|
| namespace Data |
| variable {B : StarSubalgebra ℂ A} (d : Data B) |
|
|
| theorem p_square : (d.p:A)*(d.p:A)=(d.p:A) := |
| congrArg Subtype.val d.projection.isIdempotentElem.eq |
| theorem p_star : star (d.p:A)=(d.p:A) := |
| congrArg Subtype.val d.projection.isSelfAdjoint.star_eq |
| theorem q_square : (d.q:A)*(d.q:A)=(d.q:A) := |
| congrArg Subtype.val d.projectionq.isIdempotentElem.eq |
| theorem q_star : star (d.q:A)=(d.q:A) := |
| congrArg Subtype.val d.projectionq.isSelfAdjoint.star_eq |
| theorem trace_pa (a : B) : d.τ (d.p*a)=d.τ (a*d.p) := d.tracial _ _ |
| theorem trace_pap (a : B) : d.τ (d.p*a*d.p)=d.τ (a*d.p) := by |
| rw [mul_assoc,d.tracial,mul_assoc,d.projection.isIdempotentElem.eq] |
|
|
| theorem part_part (a : B) (n m : ℕ) : |
| star (part (d.p:A) (d.q:A) d.C n)*(a:A)*part (d.p:A) (d.q:A) d.C m= |
| if n=m then d.τ (a*d.p) • (d.q:A) else 0 := by |
| calc |
| _ = (d.q:A)*(star (d.C n)*((d.p*a*d.p:B):A)*d.C m)*(d.q:A) := by |
| simp only [part,star_mul,d.p_star,d.q_star,MulMemClass.coe_mul] |
| noncomm_ring |
| _ = _ := by |
| rw [d.C_cov,d.trace_pap] |
| split_ifs <;> simp only [mul_smul_comm,smul_mul_assoc,mul_one,d.q_square,mul_zero,zero_mul] |
|
|
| theorem part_remainder (a : B) (n m : ℕ) : |
| star (part (d.p:A) (d.q:A) d.C n)*(a:A)*remainder (d.p:A) (d.q:A) d.C m=0 := by |
| calc |
| _ = (d.q:A)*(star (d.C n)*((d.p*a:B):A)*d.C m)- |
| (d.q:A)*(star (d.C n)*((d.p*a*d.p:B):A)*d.C m)*(d.q:A) := by |
| simp only [part,remainder,star_mul,d.p_star,d.q_star,MulMemClass.coe_mul] |
| noncomm_ring |
| _ = 0 := by |
| rw [d.C_cov,d.C_cov,d.trace_pa,d.trace_pap] |
| split_ifs <;> |
| simp only [mul_smul_comm,smul_mul_assoc,mul_one,d.q_square,mul_zero,zero_mul,sub_self] |
|
|
| theorem remainder_part (a : B) (n m : ℕ) : |
| star (remainder (d.p:A) (d.q:A) d.C n)*(a:A)*part (d.p:A) (d.q:A) d.C m=0 := by |
| have h := congrArg star (d.part_remainder (star a) m n) |
| simpa only [star_mul,star_star,StarMemClass.coe_star,star_zero,mul_assoc] using h |
|
|
| theorem remainder_remainder (a : B) (n m : ℕ) : |
| star (remainder (d.p:A) (d.q:A) d.C n)*(a:A)*remainder (d.p:A) (d.q:A) d.C m= |
| if n=m then d.τ a • (1:A)-d.τ (a*d.p) • (d.q:A) else 0 := by |
| calc |
| _ = star (d.C n)*(a:A)*d.C m- |
| (star (d.C n)*((a*d.p:B):A)*d.C m)*(d.q:A)- |
| (d.q:A)*(star (d.C n)*((d.p*a:B):A)*d.C m)+ |
| (d.q:A)*(star (d.C n)*((d.p*a*d.p:B):A)*d.C m)*(d.q:A) := by |
| simp only [part,remainder,star_sub,star_mul,d.p_star,d.q_star,MulMemClass.coe_mul] |
| noncomm_ring |
| _ = _ := by |
| simp only [d.C_cov,d.trace_pa,d.trace_pap] |
| split_ifs <;> |
| simp only [mul_smul_comm,smul_mul_assoc,mul_one,one_mul,d.q_square, |
| mul_zero,zero_mul,sub_zero,add_zero] |
| all_goals abel |
|
|
| theorem part_D (a : B) (n : ℕ) : |
| star (part (d.p:A) (d.q:A) d.C n)*(a:A)*d.D=0 := by |
| calc |
| _ = (d.q:A)*(star (d.C n)*((d.p*a:B):A)*d.D) := by |
| simp only [part,star_mul,d.p_star,d.q_star,MulMemClass.coe_mul,mul_assoc] |
| _ = 0 := by rw [d.C_D,mul_zero] |
|
|
| theorem D_part (a : B) (n : ℕ) : |
| star d.D*(a:A)*part (d.p:A) (d.q:A) d.C n=0 := by |
| calc |
| _ = (star d.D*((a*d.p:B):A)*d.C n)*(d.q:A) := by |
| simp only [part,MulMemClass.coe_mul,mul_assoc] |
| _ = 0 := by rw [d.D_C,zero_mul] |
|
|
| theorem remainder_D (a : B) (n : ℕ) : |
| star (remainder (d.p:A) (d.q:A) d.C n)*(a:A)*d.D=0 := by |
| simp only [remainder,star_sub,sub_mul,d.C_D,d.part_D,sub_self] |
|
|
| theorem D_remainder (a : B) (n : ℕ) : |
| star d.D*(a:A)*remainder (d.p:A) (d.q:A) d.C n=0 := by |
| simp only [remainder,mul_sub,d.D_C,d.D_part,sub_self] |
|
|
| theorem shifted_covariance (a : B) (n m : ℕ) : |
| star (shifted (d.p:A) (d.q:A) d.D d.C n)*(a:A)*shifted (d.p:A) (d.q:A) d.D d.C m= |
| if n=m then d.τ a • (1:A) else 0 := by |
| cases n with |
| | zero => |
| cases m with |
| | zero => |
| simp only [shifted,star_add,add_mul,mul_add,d.D_cov,d.D_remainder,d.remainder_D, |
| d.remainder_remainder,ite_true,add_zero,zero_add] |
| abel |
| | succ m => |
| simp only [shifted,star_add,add_mul,mul_add,d.D_part,d.D_remainder,d.remainder_part, |
| d.remainder_remainder,Nat.zero_ne_add_one,ite_false,add_zero] |
| | succ n => |
| cases m with |
| | zero => |
| simp only [shifted,star_add,add_mul,mul_add,d.part_D,d.part_remainder,d.remainder_D, |
| d.remainder_remainder,Nat.add_one_ne_zero,ite_false,add_zero] |
| | succ m => |
| simp only [shifted,star_add,add_mul,mul_add,d.part_part,d.part_remainder,d.remainder_part, |
| d.remainder_remainder,Nat.add_right_cancel_iff,add_zero,zero_add] |
| split_ifs <;> abel |
|
|
| end Data |
| end |
| variable {A : Type u_3} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A] |
|
|
| def Orthogonal (B : StarSubalgebra ℂ A) (x y : A) : Prop := |
| ∀ a : B,star x*(a:A)*y=0 |
|
|
| namespace Orthogonal |
| variable {B : StarSubalgebra ℂ A} {x y z : A} |
| theorem add_left (h : Orthogonal B x z) (k : Orthogonal B y z) : Orthogonal B (x+y) z := by |
| intro a |
| simp only [star_add,add_mul,h a,k a,add_zero] |
| theorem sub_left (h : Orthogonal B x z) (k : Orthogonal B y z) : Orthogonal B (x-y) z := by |
| intro a |
| simp only [star_sub,sub_mul,h a,k a,sub_self] |
| theorem add_right (h : Orthogonal B x y) (k : Orthogonal B x z) : Orthogonal B x (y+z) := by |
| intro a |
| simp only [mul_add,h a,k a,add_zero] |
| theorem sub_right (h : Orthogonal B x y) (k : Orthogonal B x z) : Orthogonal B x (y-z) := by |
| intro a |
| simp only [mul_sub,h a,k a,sub_self] |
| theorem symm (h : Orthogonal B x y) : Orthogonal B y x := by |
| intro a |
| have ht := congrArg star (h (star a)) |
| simpa only [star_mul,star_star,StarMemClass.coe_star,star_zero,mul_assoc] using ht |
| theorem sandwich_left (h : Orthogonal B x y) (p q : B) : |
| Orthogonal B ((p:A)*x*(q:A)) y := by |
| intro a |
| calc |
| _ = star (q:A)*(star x*((star p*a:B):A)*y) := by |
| simp only [star_mul,StarMemClass.coe_star,MulMemClass.coe_mul,mul_assoc] |
| _ = 0 := by rw [h (star p*a),mul_zero] |
| theorem sandwich_right (h : Orthogonal B x y) (p q : B) : |
| Orthogonal B x ((p:A)*y*(q:A)) := (h.symm.sandwich_left p q).symm |
|
|
| theorem part_left {C : ℕ → A} (h : ∀ n,Orthogonal B (C n) y) (p q : B) (n : ℕ) : |
| Orthogonal B (part (p:A) (q:A) C n) y := (h n).sandwich_left p q |
| theorem part_right {C : ℕ → A} (h : ∀ n,Orthogonal B x (C n)) (p q : B) (n : ℕ) : |
| Orthogonal B x (part (p:A) (q:A) C n) := (h n).sandwich_right p q |
| theorem remainder_left {C : ℕ → A} (h : ∀ n,Orthogonal B (C n) y) (p q : B) (n : ℕ) : |
| Orthogonal B (remainder (p:A) (q:A) C n) y := (h n).sub_left (part_left h p q n) |
| theorem remainder_right {C : ℕ → A} (h : ∀ n,Orthogonal B x (C n)) (p q : B) (n : ℕ) : |
| Orthogonal B x (remainder (p:A) (q:A) C n) := (h n).sub_right (part_right h p q n) |
|
|
| theorem shifted_left {C : ℕ → A} {D : A} (hD : Orthogonal B D y) |
| (hC : ∀ n,Orthogonal B (C n) y) (p q : B) (n : ℕ) : |
| Orthogonal B (shifted (p:A) (q:A) D C n) y := by |
| cases n with |
| | zero => exact hD.add_left (remainder_left hC p q 0) |
| | succ n => exact (part_left hC p q n).add_left (remainder_left hC p q (n+1)) |
| theorem shifted_right {C : ℕ → A} {D : A} (hD : Orthogonal B x D) |
| (hC : ∀ n,Orthogonal B x (C n)) (p q : B) (n : ℕ) : |
| Orthogonal B x (shifted (p:A) (q:A) D C n) := |
| (shifted_left hD.symm (fun n => (hC n).symm) p q n).symm |
|
|
| theorem shifted {C F : ℕ → A} {D E : A} (hDE : Orthogonal B D E) |
| (hDF : ∀ m,Orthogonal B D (F m)) (hCE : ∀ n,Orthogonal B (C n) E) |
| (hCF : ∀ n m,Orthogonal B (C n) (F m)) (p q r s : B) (n m : ℕ) : |
| Orthogonal B (RectangularAbsorption.shifted (p:A) (q:A) D C n) |
| (RectangularAbsorption.shifted (r:A) (s:A) E F m) := |
| shifted_left (shifted_right hDE hDF r s m) |
| (fun n => shifted_right (hCE n) (hCF n) r s m) p q n |
| end Orthogonal |
|
|
| structure Family (B : StarSubalgebra ℂ A) (I : Type u_4) where |
| data : I → Data B |
| CC : ∀ i j,i≠j → ∀ n m,Orthogonal B ((data i).C n) ((data j).C m) |
| CD : ∀ i j,i≠j → ∀ n,Orthogonal B ((data i).C n) (data j).D |
| DD : ∀ i j,i≠j → Orthogonal B (data i).D (data j).D |
|
|
| namespace Family |
| open scoped Classical |
| variable {B : StarSubalgebra ℂ A} {I : Type u_5} (d : Family B I) |
|
|
| def output (i : I) (n : ℕ) : A := |
| shifted ((d.data i).p:A) ((d.data i).q:A) (d.data i).D (d.data i).C n |
|
|
| theorem output_cross {i j : I} (hij : i≠j) (n m : ℕ) : |
| Orthogonal B (d.output i n) (d.output j m) := |
| Orthogonal.shifted (d.DD i j hij) |
| (fun m => (d.CD j i hij.symm m).symm) (d.CD i j hij) (d.CC i j hij) |
| (d.data i).p (d.data i).q (d.data j).p (d.data j).q n m |
|
|
| theorem output_covariance (i j : I) (n m : ℕ) (a : B) : |
| star (d.output i n)*(a:A)*d.output j m= |
| if i=j ∧ n=m then (d.data i).τ a • (1:A) else 0 := by |
| classical |
| by_cases hij : i=j |
| · subst j |
| simpa only [output,true_and] using (d.data i).shifted_covariance a n m |
| · rw [ite_eq_right (fun h => hij h.1)] |
| exact d.output_cross hij n m a |
|
|
| theorem output_generation (σ : I → I) |
| (hpq : ∀ i,(d.data (σ i)).p=(d.data i).q) |
| (hqp : ∀ i,(d.data (σ i)).q=(d.data i).p) |
| (M : StarSubalgebra ℂ A) |
| (hpM : ∀ i,((d.data i).p:A)∈M) (hqM : ∀ i,((d.data i).q:A)∈M) : |
| ((∀ i,paired (d.data i).D (d.data (σ i)).D∈M) ∧ |
| ∀ i n,paired ((d.data i).C n) ((d.data (σ i)).C n)∈M) ↔ |
| ∀ i n,paired (d.output i n) (d.output (σ i) n)∈M := by |
| have he (i : I) : |
| (paired (d.data i).D (d.data (σ i)).D∈M ∧ |
| ∀ n,paired ((d.data i).C n) ((d.data (σ i)).C n)∈M) ↔ |
| ∀ n,paired (d.output i n) (d.output (σ i) n)∈M := by |
| have hE : ((d.data i).q:A)*(d.data (σ i)).D*((d.data i).p:A)=(d.data (σ i)).D := by |
| simpa only [hpq,hqp] using (d.data (σ i)).D_supported |
| simpa only [output,hpq,hqp] using |
| generation_equivalence ((d.data i).p:A) ((d.data i).q:A) (d.data i).D (d.data (σ i)).D |
| (d.data i).p_square (d.data i).q_square (d.data i).p_star (d.data i).q_star |
| (d.data i).D_supported hE (d.data i).C (d.data (σ i)).C M (hpM i) (hqM i) |
| constructor |
| · rintro ⟨hD,hC⟩ i |
| exact (he i).mp ⟨hD i,hC i⟩ |
| · intro h |
| exact ⟨fun i => ((he i).mpr (h i)).1,fun i => ((he i).mpr (h i)).2⟩ |
|
|
| end Family |
| end FreeGroupFactorMain.RectangularAbsorption |
|
|
| end |
|
|
| end OAI |
|
|
