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refactor: Lint

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jstoobysmith 2024-10-09 15:20:23 +00:00
parent a39aeeed8b
commit 4054665c38
3 changed files with 109 additions and 75 deletions
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56
HepLean/Mathematics/PiTensorProduct.lean
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@ -33,8 +33,7 @@ open TensorProduct
## induction principals for pi tensor products
-/
lemma tensorProd_piTensorProd_ext
{f g : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) →ₗ[R] M}
lemma induction_tmul {f g : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) →ₗ[R] M}
(h : ∀ p q, f (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)
= g (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)) : f = g := by
apply TensorProduct.ext'
@ -56,8 +55,10 @@ lemma tensorProd_piTensorProd_ext
lemma induction_assoc
{f g : ((⨂[R] i : ι1, s1 i) ⊗[R] (⨂[R] i : ι2, s2 i) ⊗[R] ⨂[R] i : ι3, s3 i) →ₗ[R] M}
(h : ∀ p q m, f (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q ⊗ₜ[R] PiTensorProduct.tprod R m)
= g (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q ⊗ₜ[R] PiTensorProduct.tprod R m)) : f = g := by
(h : ∀ p q m, f (PiTensorProduct.tprod R p ⊗ₜ[R]
PiTensorProduct.tprod R q ⊗ₜ[R] PiTensorProduct.tprod R m)
= g (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q
⊗ₜ[R] PiTensorProduct.tprod R m)) : f = g := by
apply TensorProduct.ext'
refine fun x ↦
PiTensorProduct.induction_on' x ?_ (by
@ -65,7 +66,7 @@ lemma induction_assoc
simp [map_add, add_tmul, hx, hy])
intro rx fx
intro y
simp
simp only [PiTensorProduct.tprodCoeff_eq_smul_tprod]
simp only [smul_tmul, tmul_smul, LinearMapClass.map_smul]
apply congrArg
let f' : ((⨂[R] i : ι2, s2 i) ⊗[R] ⨂[R] i : ι3, s3 i) →ₗ[R] M := {
@ -83,14 +84,15 @@ lemma induction_assoc
change f' y = g' y
apply congrFun
refine DFunLike.coe_fn_eq.mpr ?H.h.h.a
apply tensorProd_piTensorProd_ext
apply induction_tmul
intro p q
exact h fx p q
lemma induction_assoc'
{f g : (((⨂[R] i : ι1, s1 i) ⊗[R] (⨂[R] i : ι2, s2 i)) ⊗[R] ⨂[R] i : ι3, s3 i) →ₗ[R] M}
(h : ∀ p q m, f ((PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q) ⊗ₜ[R] PiTensorProduct.tprod R m)
= g ((PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q) ⊗ₜ[R] PiTensorProduct.tprod R m)) : f = g := by
(h : ∀ p q m, f ((PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q) ⊗ₜ[R]
PiTensorProduct.tprod R m) = g ((PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)
⊗ₜ[R] PiTensorProduct.tprod R m)) : f = g := by
apply TensorProduct.ext'
intro x
refine fun y ↦
@ -98,7 +100,7 @@ lemma induction_assoc'
intro a b hy hx
simp [map_add, add_tmul, tmul_add, hy, hx])
intro ry fy
simp
simp only [PiTensorProduct.tprodCoeff_eq_smul_tprod, tmul_smul, map_smul]
apply congrArg
let f' : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) →ₗ[R] M := {
toFun := fun y => f (y ⊗ₜ[R] PiTensorProduct.tprod R fy),
@ -115,13 +117,14 @@ lemma induction_assoc'
change f' x = g' x
apply congrFun
refine DFunLike.coe_fn_eq.mpr ?H.h.h.a
apply tensorProd_piTensorProd_ext
apply induction_tmul
intro p q
exact h p q fy
lemma induction_tmul_mod
{f g : ((⨂[R] i : ι1, s1 i) ⊗[R] N) →ₗ[R] M}
(h : ∀ p m, f (PiTensorProduct.tprod R p ⊗ₜ[R] m) = g (PiTensorProduct.tprod R p ⊗ₜ[R] m)) : f = g := by
(h : ∀ p m, f (PiTensorProduct.tprod R p ⊗ₜ[R] m) = g (PiTensorProduct.tprod R p ⊗ₜ[R] m)) :
f = g := by
apply TensorProduct.ext'
refine fun y ↦
PiTensorProduct.induction_on' y ?_ (by
@ -135,7 +138,8 @@ lemma induction_tmul_mod
lemma induction_mod_tmul
{f g : (N ⊗[R] ⨂[R] i : ι1, s1 i) →ₗ[R] M}
(h : ∀ m p, f (m ⊗ₜ[R] PiTensorProduct.tprod R p) = g (m ⊗ₜ[R] PiTensorProduct.tprod R p)) : f = g := by
(h : ∀ m p, f (m ⊗ₜ[R] PiTensorProduct.tprod R p) = g (m ⊗ₜ[R] PiTensorProduct.tprod R p)) :
f = g := by
apply TensorProduct.ext'
intro x
refine fun y ↦
@ -162,15 +166,20 @@ instance : (i : ι1 ⊕ ι2) → Module R ((fun i => Sum.elim s1 s2 i) i) := fun
| Sum.inr i => inst2' i
/-- Takes a map `(i : ι1 ⊕ ι2) → Sum.elim s1 s2 i` to the underlying map `(i : ι1) → s1 i `. -/
private def pureInl (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) : (i : ι1) → s1 i :=
fun i => f (Sum.inl i)
/-- Takes a map `(i : ι1 ⊕ ι2) → Sum.elim s1 s2 i` to the underlying map `(i : ι2) → s2 i `. -/
private def pureInr (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) : (i : ι2) → s2 i :=
fun i => f (Sum.inr i)
section
variable [DecidableEq (ι1 ⊕ ι2)] [DecidableEq ι1] [DecidableEq ι2]
lemma pureInl_update_left (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι1)
variable [DecidableEq (ι1 ⊕ ι2)]
omit inst1 inst2
lemma pureInl_update_left [DecidableEq ι1] (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι1)
(v1 : s1 x) : pureInl (Function.update f (Sum.inl x) v1) =
Function.update (pureInl f) x v1 := by
funext y
@ -187,7 +196,7 @@ lemma pureInr_update_left (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι1)
funext y
simp [pureInr, Function.update, Sum.inl.injEq, Sum.elim_inl]
lemma pureInr_update_right (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι2)
lemma pureInr_update_right [DecidableEq ι2] (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι2)
(v2 : s2 x) : pureInr (Function.update f (Sum.inr x) v2) =
Function.update (pureInr f) x v2 := by
funext y
@ -205,7 +214,11 @@ lemma pureInl_update_right (f : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i) (x : ι2
simp [pureInl, Function.update, Sum.inr.injEq, Sum.elim_inr]
end
def domCoprod : MultilinearMap R (Sum.elim s1 s2) ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) where
/-- The multilinear map from `(Sum.elim s1 s2)` to `((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i)`
defined by splitting elements of `(Sum.elim s1 s2)` into two parts. -/
def domCoprod :
MultilinearMap R (Sum.elim s1 s2) ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) where
toFun f := (PiTensorProduct.tprod R (pureInl f)) ⊗ₜ
(PiTensorProduct.tprod R (pureInr f))
map_add' f xy v1 v2 := by
@ -235,10 +248,12 @@ def domCoprod : MultilinearMap R (Sum.elim s1 s2) ((⨂[R] i : ι1, s1 i) ⊗[R]
simp only [Sum.elim_inr, pureInl_update_right, pureInr_update_right, MultilinearMap.map_smul,
tmul_smul]
/-- Expand `PiTensorProduct` on sums into a `TensorProduct` of two factors. -/
def tmulSymm : (⨂[R] i : ι1 ⊕ ι2, (Sum.elim s1 s2) i) →ₗ[R]
((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) := PiTensorProduct.lift domCoprod
/-- Produces a map `(i : ι1 ⊕ ι2) → Sum.elim s1 s2 i` from a map `(i : ι1) → s1 i` and a
map `q : (i : ι2) → s2 i`. -/
def elimPureTensor (p : (i : ι1) → s1 i) (q : (i : ι2) → s2 i) : (i : ι1 ⊕ ι2) → Sum.elim s1 s2 i :=
fun x =>
match x with
@ -248,6 +263,7 @@ def elimPureTensor (p : (i : ι1) → s1 i) (q : (i : ι2) → s2 i) : (i : ι1
section
variable [DecidableEq ι1] [DecidableEq ι2]
omit inst1 inst2
lemma elimPureTensor_update_right (p : (i : ι1) → s1 i) (q : (i : ι2) → s2 i)
(y : ι2) (r : s2 y) : elimPureTensor p (Function.update q y r) =
@ -286,6 +302,8 @@ lemma elimPureTensor_update_left (p : (i : ι1) → s1 i) (q : (i : ι2) → s2
end
/-- The multilinear map valued in multilinear maps defined by combining
`(i : ι1) → s1 i` and `q : (i : ι2) → s2 i` into a PiTensorProduct. -/
def elimPureTensorMulLin : MultilinearMap R s1
(MultilinearMap R s2 (⨂[R] i : ι1 ⊕ ι2, (Sum.elim s1 s2) i)) where
toFun p := {
@ -311,6 +329,7 @@ def elimPureTensorMulLin : MultilinearMap R s1
intro y
simp
/-- Collapse a `TensorProduct` of `PiTensorProduct` into a `PiTensorProduct`. -/
def tmul : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) →ₗ[R]
⨂[R] i : ι1 ⊕ ι2, (Sum.elim s1 s2) i := TensorProduct.lift {
toFun := fun a ↦
@ -319,6 +338,7 @@ def tmul : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) →ₗ[R]
map_add' := fun a b ↦ by simp
map_smul' := fun r a ↦ by simp}
/-- THe equivalence formed by combining a `TensorProduct` into a `PiTensorProduct`. -/
def tmulEquiv : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) ≃ₗ[R]
⨂[R] i : ι1 ⊕ ι2, (Sum.elim s1 s2) i :=
LinearEquiv.ofLinear tmul tmulSymm
@ -336,7 +356,7 @@ def tmulEquiv : ((⨂[R] i : ι1, s1 i) ⊗[R] ⨂[R] i : ι2, s2 i) ≃ₗ[R]
| Sum.inl x => rfl
| Sum.inr x => rfl)
(by
apply tensorProd_piTensorProd_ext
apply induction_tmul
intro p q
simp only [tmulSymm, domCoprod, tmul, elimPureTensorMulLin, LinearMap.coe_comp,
Function.comp_apply, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk, PiTensorProduct.lift.tprod,