File size: 5,274 Bytes
216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockModules
import OAI.Algebra.Finitistic.Ordinary.OrdinaryBalancedCongruence
namespace OAI
noncomputable section
namespace LittleFinitistic.OrdinarySimulation.FirstBlockTensor
universe u
variable (k L R O T Z : Type u) [Field k] [Ring L] [Ring R] [Algebra k L] [Algebra k R]
variable [AddCommGroup O] [Module k O] [Module R O] [Module Lᵐᵒᵖ O]
[IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O] [SMulCommClass R Lᵐᵒᵖ O]
variable [AddCommGroup T] [Module k T] [Module L T] [Module Rᵐᵒᵖ T]
[IsScalarTower k L T] [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T]
variable [AddCommGroup Z] [Module k Z] [Module L Z] [IsScalarTower k L Z]
abbrev source := LeftTensor k (L × R) (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)
abbrev target := BlockModule.Second k L R (LeftTensor k L R O Z)
def forward : source k L R O T Z →ₗ[k] target k L R O Z :=
BalancedTensor.lift ((BalancedTensor.mk k L O Z).compl₁₂
(show OffDiagonal O T →ₗ[k] O from LinearMap.fst k O T)
(show BlockModule.First k L R Z →ₗ[k] Z from LinearMap.id)) (by
intro d x z
exact BalancedTensor.balance (k := k) (A := L) (M := O) (N := Z) d.1 x.1 (show Z from z))
def inverse : target k L R O Z →ₗ[k] source k L R O T Z :=
BalancedTensor.lift ((BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)).compl₁₂
(show O →ₗ[k] OffDiagonal O T from LinearMap.inl k O T)
(show Z →ₗ[k] BlockModule.First k L R Z from LinearMap.id)) (by
intro a o z
have h := BalancedTensor.balance (k := k) (A := L × R) (M := OffDiagonal O T)
(N := BlockModule.First k L R Z) (a,0) (o,0) z
change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)
(MulOpposite.op a • o,(0:Rᵐᵒᵖ) • (0:T)) (show BlockModule.First k L R Z from z) =
BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (o,0)
(show BlockModule.First k L R Z from a • z) at h
rw [smul_zero] at h
exact h)
omit [Algebra k L] [Algebra k R] [Module R O] [IsScalarTower k R O]
[IsScalarTower k Lᵐᵒᵖ O] [SMulCommClass R Lᵐᵒᵖ O] [Module L T]
[IsScalarTower k L T] [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T]
[IsScalarTower k L Z] in
theorem other_block_zero (t : T) (z : BlockModule.First k L R Z) :
BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (0,t) z = 0 := by
have h := BalancedTensor.balance (k := k) (A := L × R) (M := OffDiagonal O T)
(N := BlockModule.First k L R Z) (0,1) (0,t) z
change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)
((0:Lᵐᵒᵖ) • (0:O),(1:Rᵐᵒᵖ) • t) z =
BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (0,t) ((0:L) • (show Z from z)) at h
erw [zero_smul, one_smul, zero_smul, map_zero] at h
exact h
omit [Module R O] [IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O]
[SMulCommClass R Lᵐᵒᵖ O] [Module L T] [IsScalarTower k L T]
[IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] [IsScalarTower k L Z] in
theorem inverse_forward (x : source k L R O T Z) :
inverse k L R O T Z (forward k L R O T Z x) = x := by
let F := (inverse k L R O T Z).comp (forward k L R O T Z)
have h : F = LinearMap.id := by
apply BalancedTensor.ext
intro x z
change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (x.1,0) z =
BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) x z
let q := (BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)).flip z
have hx : x = (x.1,0) + (0,x.2) := Prod.ext (add_zero _).symm (zero_add _).symm
exact ((congrArg q hx).trans ((q.map_add _ _).trans
((congrArg (q (x.1,0) + ·) (other_block_zero k L R O T Z x.2 z)).trans (add_zero _)))).symm
exact congrArg (fun f => f x) h
omit [Module R O] [IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O]
[SMulCommClass R Lᵐᵒᵖ O] [Module L T] [IsScalarTower k L T]
[IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] [IsScalarTower k L Z] in
theorem forward_inverse (x : target k L R O Z) :
forward k L R O T Z (inverse k L R O T Z x) = x := by
have h : (forward k L R O T Z).comp (inverse k L R O T Z) = LinearMap.id := by
apply BalancedTensor.ext
intro o z
rfl
exact congrArg (fun f => f x) h
/-- Tensor on the first block selects O and has its actual output on the second block. -/
def equivalence : source k L R O T Z ≃ₗ[L × R] target k L R O Z where
toFun := forward k L R O T Z
invFun := inverse k L R O T Z
left_inv := inverse_forward k L R O T Z
right_inv := forward_inverse k L R O T Z
map_add' := (forward k L R O T Z).map_add
map_smul' d x := by
let F := forward k L R O T Z
have h : F.comp (LeftTensor.action k (L × R) (L × R) (OffDiagonal O T)
(BlockModule.First k L R Z) d) =
(Algebra.lsmul k k (target k L R O Z) d).comp F := by
apply BalancedTensor.ext
intro x z
rfl
exact congrArg (fun f => f x) h
end LittleFinitistic.OrdinarySimulation.FirstBlockTensor
end
end OAI
|