feat(LinearAlgebra/UnitaryGroup): Add properties of Special Unitary Group #12799
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chrisflav
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May 19, 2024
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| theorem star_mem_iff {U : Matrix n n α} : | ||
| star U ∈ specialUnitaryGroup n α ↔ U ∈ specialUnitaryGroup n α := | ||
| ⟨fun h => star_star U ▸ star_mem h, star_mem⟩ | ||
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| instance : Star (specialUnitaryGroup n α) := | ||
| ⟨fun U => ⟨star U, star_mem U.prop⟩⟩ |
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Suggested change
| theorem star_mem_iff {U : Matrix n n α} : | |
| star U ∈ specialUnitaryGroup n α ↔ U ∈ specialUnitaryGroup n α := | |
| ⟨fun h => star_star U ▸ star_mem h, star_mem⟩ | |
| instance : Star (specialUnitaryGroup n α) := | |
| ⟨fun U => ⟨star U, star_mem U.prop⟩⟩ | |
| theorem star_mem_iff {U : Matrix n n α} : | |
| star U ∈ specialUnitaryGroup n α ↔ U ∈ specialUnitaryGroup n α := | |
| ⟨fun h ↦ star_star U ▸ star_mem h, star_mem⟩ | |
| instance : Star (specialUnitaryGroup n α) := | |
| ⟨fun U ↦ ⟨star U, star_mem U.prop⟩⟩ |
I think this is the preferred style.
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| instance : Group (specialUnitaryGroup n α) := | ||
| { Submonoid.toMonoid _ with | ||
| inv := star | ||
| mul_left_inv := star_mul_self } |
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Suggested change
| instance : Group (specialUnitaryGroup n α) := | |
| { Submonoid.toMonoid _ with | |
| inv := star | |
| mul_left_inv := star_mul_self } | |
| instance : Group (specialUnitaryGroup n α) where | |
| inv := star | |
| mul_left_inv := star_mul_self |
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| instance : InvolutiveStar (specialUnitaryGroup n α) := | ||
| ⟨by | ||
| intro x | ||
| ext | ||
| rw [coe_star, coe_star, star_star]⟩ |
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Suggested change
| instance : InvolutiveStar (specialUnitaryGroup n α) := | |
| ⟨by | |
| intro x | |
| ext | |
| rw [coe_star, coe_star, star_star]⟩ | |
| instance : InvolutiveStar (specialUnitaryGroup n α) := | |
| star_involutive x := by | |
| ext | |
| rw [coe_star, coe_star, star_star] |
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| instance : StarMul (specialUnitaryGroup n α) := | ||
| ⟨by | ||
| intro x y | ||
| ext | ||
| rw [coe_star, Submonoid.coe_mul, Submonoid.coe_mul, coe_star, coe_star, star_mul]⟩ |
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Suggested change
| instance : StarMul (specialUnitaryGroup n α) := | |
| ⟨by | |
| intro x y | |
| ext | |
| rw [coe_star, Submonoid.coe_mul, Submonoid.coe_mul, coe_star, coe_star, star_mul]⟩ | |
| instance : StarMul (specialUnitaryGroup n α) := | |
| star_mul x y := by | |
| ext | |
| rw [coe_star, Submonoid.coe_mul, Submonoid.coe_mul, coe_star, coe_star, star_mul] |
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| @[simp] | ||
| theorem star_mul_self (U : specialUnitaryGroup n α) : star U * U = 1 := | ||
| Subtype.ext <| coe_star_mul_self U | ||
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| @[simp] | ||
| theorem mul_star_self (U : specialUnitaryGroup n α) : U * star U = 1 := | ||
| Subtype.ext <| coe_mul_star_self U | ||
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Currently
example (U : specialUnitaryGroup n α) : U * star U = 1 := by
simpright after these lemmas fails because simp is running out of heartbeats. Changing abbrev specialUnitaryGroup to def specialUnitaryGroup fixes this. Maybe the other abbrevs in this file should also be defs?
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| theorem star_eq_inv (U : specialUnitaryGroup n α) : star U = U⁻¹ := | ||
| rfl | ||
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| theorem star_eq_inv' : (star : specialUnitaryGroup n α → specialUnitaryGroup n α) = Inv.inv := | ||
| rfl | ||
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Currently this
example (U : unitaryGroup n α) : U⁻¹ = (star U : Matrix n n α) := by
simpworks, but
example (U : specialUnitaryGroup n α) : U⁻¹ = (star U : Matrix n n α) := by
simpdoes not. I am not sure what the correct behavior should be, but it should probably be consistent.
chrisflav
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This PR looks mostly fine, but there are still some places where it needs some work. Are you planning to continue working on this PR? Would you like some help from others, or do you want to hand it over completely? (In the latter case, please label it with |
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Add properties of the special unitary group, mirroring the properties of found in Algebra/Star/Unitary.lean. In particular, I add an instance of
specialUnitaryGroupas aGroup,Star,InvolutiveStar, andStarMul.