| import OAI.Analysis.InterpolatedFactors.CornerGaussian |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 |
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| noncomputable section |
| namespace FreeGroupFactorMain.CircularRotation |
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| def q : ℂ := (Real.sqrt 2 : ℂ)⁻¹ |
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| theorem q_star : star q=q := by |
| simp only [q,Complex.star_def,map_inv₀,Complex.conj_ofReal] |
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| theorem q_sq : q^2=1/2 := by |
| have hs : (Real.sqrt 2 : ℂ)^2=2 := by |
| exact_mod_cast Real.sq_sqrt (by norm_num : (0:ℝ)≤2) |
| rw [q,inv_pow,hs] |
| norm_num |
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| theorem two_q_mul {A : Type u_1} [AddCommGroup A] [Module ℂ A] (X : A) : |
| q • (q • X+q • X)=X := by |
| rw [smul_add,smul_smul,← add_smul,← pow_two,q_sq] |
| norm_num |
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| theorem qI_star : star (q*Complex.I)=-(q*Complex.I) := by |
| rw [star_mul,q_star,Complex.star_def,Complex.conj_I] |
| ring |
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| section |
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| variable {A : Type u_2} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A] |
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| def plus (X Y : A) : A := q • X+(q*Complex.I) • Y |
| def minus (X Y : A) : A := q • X-(q*Complex.I) • Y |
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| omit [StarRing A] [StarModule ℂ A] in |
| theorem plus_add_minus (X Y : A) : q • (plus X Y+minus X Y)=X := by |
| rw [plus,minus] |
| have he : q • X+(q*Complex.I) • Y+(q • X-(q*Complex.I) • Y)=q • X+q • X := by abel |
| rw [he,two_q_mul] |
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| omit [StarRing A] [StarModule ℂ A] in |
| theorem plus_sub_minus (X Y : A) : (-q*Complex.I) • (plus X Y-minus X Y)=Y := by |
| have he : plus X Y-minus X Y=(q*Complex.I) • Y+(q*Complex.I) • Y := by unfold plus minus;abel |
| rw [he,smul_add,smul_smul,← add_smul] |
| have hc : (-q*Complex.I)*(q*Complex.I)+(-q*Complex.I)*(q*Complex.I)=1 := by |
| calc |
| _ = -2*q^2*Complex.I^2 := by ring |
| _ = 1 := by rw [q_sq,Complex.I_sq];norm_num |
| rw [hc,one_smul] |
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| theorem paired_plus (X Y : A) : plus X Y+star (minus X Y)=plus (X+star X) (Y+star Y) := by |
| simp only [plus,minus,star_sub,star_smul,q_star,qI_star,neg_smul,smul_add] |
| abel |
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| theorem paired_minus (X Y : A) : minus X Y+star (plus X Y)=minus (X+star X) (Y+star Y) := by |
| simp only [plus,minus,star_add,star_smul,q_star,qI_star,neg_smul,smul_add] |
| abel |
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| theorem generation (X Y : A) (M : StarSubalgebra ℂ A) : |
| (X+star X∈M ∧ Y+star Y∈M) ↔ |
| (plus X Y+star (minus X Y)∈M ∧ minus X Y+star (plus X Y)∈M) := by |
| rw [paired_plus,paired_minus] |
| constructor |
| · rintro ⟨hX,hY⟩ |
| exact ⟨add_mem (M.smul_mem hX _) (M.smul_mem hY _),sub_mem (M.smul_mem hX _) (M.smul_mem hY _)⟩ |
| · rintro ⟨hP,hM⟩ |
| constructor |
| · rw [← plus_add_minus (X+star X) (Y+star Y)] |
| exact M.smul_mem (add_mem hP hM) _ |
| · rw [← plus_sub_minus (X+star X) (Y+star Y)] |
| exact M.smul_mem (sub_mem hP hM) _ |
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| theorem contraction (X Y a Z : A) |
| (hXX : star X*a*X=Z) (hYY : star Y*a*Y=Z) |
| (hXY : star X*a*Y=0) (hYX : star Y*a*X=0) : |
| star (plus X Y)*a*plus X Y=Z ∧ star (minus X Y)*a*minus X Y=Z ∧ |
| star (plus X Y)*a*minus X Y=0 ∧ star (minus X Y)*a*plus X Y=0 := by |
| have h1 : q*q=1/2 := by rw [← pow_two,q_sq] |
| have h2 : (q*Complex.I)*(q*Complex.I)=-(1/2) := by |
| calc |
| _ = q^2*Complex.I^2 := by ring |
| _ = -(1/2) := by rw [q_sq,Complex.I_sq];ring |
| simp only [plus,minus,star_add,star_sub,star_smul,q_star,qI_star,add_mul,sub_mul,mul_add,mul_sub, |
| smul_mul_assoc,mul_smul_comm] |
| simp only [hXX,hYY,hXY,hYX,smul_zero,add_zero,zero_add,sub_zero,zero_sub] |
| simp only [smul_smul,mul_neg,neg_neg,h1,h2] |
| repeat' constructor |
| all_goals match_scalars |
| all_goals ring_nf |
| all_goals norm_num [q_sq,Complex.I_sq] |
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| end |
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| variable {A : Type u_3} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A] |
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| def realFirst (X Y : A) : A := q • (X+Y) |
| def realSecond (X Y : A) : A := (-q*Complex.I) • (X-Y) |
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| theorem inverse_paired (X Y : A) : |
| realFirst X Y+star (realFirst X Y)=q • ((X+star Y)+(Y+star X)) ∧ |
| realSecond X Y+star (realSecond X Y)=(-q*Complex.I) • ((X+star Y)-(Y+star X)) := by |
| constructor |
| · simp only [realFirst,star_smul,q_star,star_add,smul_add] |
| abel |
| · have hs : star (-q*Complex.I)=q*Complex.I := by |
| rw [star_mul,star_neg,q_star,Complex.star_def,Complex.conj_I] |
| ring |
| simp only [realSecond,star_smul,hs,star_sub] |
| simp only [smul_sub,neg_mul,neg_smul,smul_add] |
| abel |
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| theorem inverse_contraction (X Y a Z : A) |
| (hXX : star X*a*X=Z) (hYY : star Y*a*Y=Z) |
| (hXY : star X*a*Y=0) (hYX : star Y*a*X=0) : |
| star (realFirst X Y)*a*realFirst X Y=Z ∧ star (realSecond X Y)*a*realSecond X Y=Z ∧ |
| star (realFirst X Y)*a*realSecond X Y=0 ∧ star (realSecond X Y)*a*realFirst X Y=0 := by |
| have hs : star (-q*Complex.I)=q*Complex.I := by |
| rw [star_mul,star_neg,q_star,Complex.star_def,Complex.conj_I] |
| ring |
| simp only [realFirst,realSecond,star_smul,star_add,star_sub,q_star,hs, |
| add_mul,sub_mul,mul_add,mul_sub,smul_mul_assoc,mul_smul_comm] |
| simp only [hXX,hYY,hXY,hYX,add_zero,zero_add,sub_zero,zero_sub] |
| repeat' constructor |
| all_goals match_scalars |
| all_goals ring_nf |
| all_goals norm_num [q_sq,Complex.I_sq] |
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| variable {S : Type u_4} {P : Type u_5} |
| abbrev PairedIndex (S : Type u_6) (P : Type u_7) := S ⊕ (P×Bool) |
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| def flip : PairedIndex S P → PairedIndex S P |
| | .inl i => .inl i |
| | .inr (j,b) => .inr (j,!b) |
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| def rotate (R : PairedIndex S P → A) : PairedIndex S P → A |
| | .inl i => R (.inl i) |
| | .inr (j,false) => plus (R (.inr (j,false))) (R (.inr (j,true))) |
| | .inr (j,true) => minus (R (.inr (j,false))) (R (.inr (j,true))) |
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| theorem rotate_covariance [DecidableEq S] [DecidableEq P] (R : PairedIndex S P → A) (a Z : A) |
| (h : ∀ i j,star (R i)*a*R j=if i=j then Z else 0) (i j : PairedIndex S P) : |
| star (rotate R i)*a*rotate R j=if i=j then Z else 0 := by |
| rcases i with i | ⟨i,b⟩ <;> rcases j with j | ⟨j,c⟩ |
| · exact h (.inl i) (.inl j) |
| · cases c <;> simp [rotate,plus,minus,mul_add,mul_sub,h] |
| · cases b <;> simp [rotate,plus,minus,star_add,star_sub,star_smul,add_mul,h] |
| · by_cases hij : i=j |
| · subst j |
| have hXX : star (R (.inr (i,false)))*a*R (.inr (i,false))=Z := by simpa using h (.inr (i,false)) (.inr (i,false)) |
| have hYY : star (R (.inr (i,true)))*a*R (.inr (i,true))=Z := by simpa using h (.inr (i,true)) (.inr (i,true)) |
| have hXY : star (R (.inr (i,false)))*a*R (.inr (i,true))=0 := by simpa using h (.inr (i,false)) (.inr (i,true)) |
| have hYX : star (R (.inr (i,true)))*a*R (.inr (i,false))=0 := by simpa using h (.inr (i,true)) (.inr (i,false)) |
| have hh := contraction _ _ a Z hXX hYY hXY hYX |
| cases b <;> cases c |
| · simpa only [rotate,ite_true] using hh.1 |
| · simpa [rotate] using hh.2.2.1 |
| · simpa [rotate] using hh.2.2.2 |
| · simpa only [rotate,ite_true] using hh.2.1 |
| · cases b <;> cases c <;> |
| simp [rotate,plus,minus,star_add,star_sub,star_smul,add_mul,mul_add,mul_sub,h,hij] |
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| theorem rotate_generation (R : PairedIndex S P → A) (M : StarSubalgebra ℂ A) : |
| (∀ i,R i+star (R i)∈M) ↔ (∀ i,rotate R i+star (rotate R (flip i))∈M) := by |
| constructor |
| · intro h i |
| rcases i with i | ⟨j,(_|_)⟩ |
| · exact h (.inl i) |
| · exact ((generation _ _ M).mp ⟨h (.inr (j,false)),h (.inr (j,true))⟩).1 |
| · exact ((generation _ _ M).mp ⟨h (.inr (j,false)),h (.inr (j,true))⟩).2 |
| · intro h i |
| rcases i with i | ⟨j,(_|_)⟩ |
| · exact h (.inl i) |
| · exact ((generation _ _ M).mpr ⟨h (.inr (j,false)),h (.inr (j,true))⟩).1 |
| · exact ((generation _ _ M).mpr ⟨h (.inr (j,false)),h (.inr (j,true))⟩).2 |
| end FreeGroupFactorMain.CircularRotation |
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| end |
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| end OAI |
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